PHOENIX — From his desk in a downtown workshop, Greg Hankerson is at war with a Chinese company half a world away.

Mr. Hankerson and his wife, Sim, own Vintage Industrial, which designs and makes antique-style tables, cabinets and other furniture. The 25-employee start-up produces everything at its Phoenix factory, much of it by hand.

But that hasn’t protected Mr. Hankerson from counterfeiters, who peddle cheap copies of his creations on internet marketplaces run by Alibaba, China’s largest e-commerce company. He can find hundreds of suspected counterfeits of his furniture on Alibaba’s various sites, including Taobao, a free-for-all shopping platform on which the Chinese hawk items as varied as T-shirts and televisions.

One recent day, Mr. Hankerson fired up his web browser to scan for counterfeits. Several Taobao shops sell copies of various Vintage Industrial tables, including one with A-shaped legs and another with a glass top and propeller-like base, as well as cabinets and a metal locker.

“It just keeps going and going and going,” Mr. Hankerson, 45, said. “It’s like trying to pick weeds on a 70-acre farm.”

Alibaba’s founder, Jack Ma, has pledged to enhance the fortunes of small enterprises around the world by using e-commerce to expand their global trade. In January, Mr. Ma, who is one of China’s richest people, pledged to Donald J. Trump, then the president-elect, that Alibaba would create one million jobs in the United States by connecting small businesses like Mr. Hankerson’s to increasingly wealthy Chinese shoppers.

But Mr. Hankerson wants to know why Alibaba doesn’t do more to defeat counterfeiting, which damages the very same small businesses Mr. Ma says he intends to aid.

Mr. Ma “is making himself look like someone you can trust,” Mr. Hankerson said. But, he added, his company is selling counterfeit products.

Alibaba is a company valued at $260 billion, with hundreds of millions of buyers using its sales platforms. Millions of Chinese shop there for things as varied as snacks and knickknacks and phone charging cables, while global brands flock to its high-end sales platform, Tmall. In China, Alibaba is a kingmaker for hopeful tech start-ups, while an affiliated online mobile payment system is the envy of Silicon Valley. It has a growing cloud-computing business and increasingly extols its prowess with big data.

That leaves many businesses, big and small, wondering why it has such a hard time finding fakes on its sites. Its online system for reporting counterfeits, many say, is cumbersome and prone to hiccups. Against one of the world’s biggest tech firms, Mr. Hankerson employs his iMac, $74-a-month image-searching software, his phone and a lot of time — sometimes, he says, 12 hours a day.

Yet if he doesn’t invest the effort, he worries that a flood of cut-rate replicas could undermine the future of his business. Shops on Alibaba — shops that sometimes sell outside China — offer their copies at a fraction of Mr. Hankerson’s prices. One Taobao store sold a version of his A-frame table, with a starting price of $5,295, for $24.

“If this gets out of hand, people are going to be able to buy our stuff for next for nothing,” he said. “It could be devastating for us.”

Fighting Fakes

Alibaba has long faced accusations that its sales platforms are a haven for fakes, and big organizations have been effective at bringing the problem to the fore.

Following complaints from industry groups, the office of the United States Trade Representative last year added Taobao to its list of “notorious markets” for counterfeit goods, after removing it four years earlier. The corporate owner of the Gucci and Yves Saint Laurent brand names, Kering, sued Alibaba two years ago over the prevalence of fakes on its services. Alibaba, which is fighting the suit in a New York federal court, says it has no basis.

In a report submitted to the U.S.T.R. in October, Alibaba boasted about the technology and resources it uses to clean counterfeits from its platforms. The company says its systems are powerful enough to scan 10 million product listings each day. As a result, Alibaba said, it proactively removed 380 million suspect product listings in one 12-month period.

“There are few companies (indeed, none, to our knowledge) that have taken the combination of steps Alibaba has taken, and has concrete plans to take, and certainly none that have put in place measures on the scale of Alibaba’s,” the company told the U.S.T.R.

Alibaba also argues that many of the complaints against it are unfair. Counterfeiting and piracy are widespread in the Chinese economy. As private businesspeople, Mr. Ma and his team cannot shut down the culprits producing fake goods.

Additionally, Alibaba suggests that it has been made into a scapegoat. In a statement in December, the company protested that the U.S.T.R. decision “leads us to question whether the U.S.T.R. acted based on the actual facts or was influenced by the current political climate” — presumably referring to heightened anti-China sentiment amid last year’s United States presidential election.

The U.S.T.R. and industry experts contend that there is a lot more Alibaba could and should be doing to tackle fakes.

“They are a leading technology company,” said Stephen Lamar, executive vice president at the American Apparel & Footwear Association, which represents many brands hurt by counterfeiting. “It is our hope that they use this technology to develop solutions and make sure those solutions are available for everybody.”

The U.S.T.R. notorious-markets report cited the extra challenges that small and medium enterprises face in dealing with fakes on Taobao. Smaller firms have a harder time qualifying for Alibaba’s streamlined program for removing fakes, the report said, which means small firms often encounter more bureaucracy and longer response times than some large ones. The report said that businesses regularly experience problems with Alibaba’s reporting system, including persistent error messages that stall submissions.

For small enterprises, fighting fakes on Alibaba’s sites “can become expensive, it can become frustrating, it can take time away from your sales, marketing and other creative endeavors,” Mr. Lamar said.

An Alibaba spokesman said that suggestions that small businesses do not get its attention are “false” and that they can qualify for the streamlined process if their submissions prove reliable. He added that its process for removing suspect listings is in place to address the large number of fraudulent claims the firm receives.

“There are places that our systems can be improved to make them more effective, efficient and user-friendly,” he said, “and we are working hard every day to make these improvements a reality.”

Thinking Small

Michelle Stennett, 38, a jewelry maker in Suwanee, Ga., found out counterfeiters had stolen her designs during a trip to Yiwu in eastern China in 2012 to buy supplies. Upon returning home, Ms. Stennett scoured — an Alibaba platform that sells around the world — and discovered shop after shop selling copies of her wares.

She first tried contacting the stores directly, ordering them to take down the offending goods. Some complied, but others didn’t. Next she filed a couple of complaints to Alibaba, asking the firm to remove the copies. Although those were successful, she quickly realized that the scale of the problem and the paperwork Alibaba required were too much for her to handle as a one-person business.

“There was no way I could go through the process,” she said. There were “too many.”

Despondent, Ms. Stennett said she stopped making jewelry for nearly a year. “If someone is going to take my design and mass-produce it, it is hard for me to compete with them,” she said. When she resumed, in 2013, she decided to source her components only from American suppliers, hoping to make it harder for counterfeiters to replicate her products.

Now Ms. Stennett rarely checks Alibaba, afraid of what she’ll find. After a quick search of one day in March, she spotted an image of one of her pendants, which she reported to Alibaba. It is no longer on the platform, but she said she still does not have time to hunt for fakes.

“You’re losing out on money,” she said, “because time is money.”

Alibaba did not respond to a request for comment on Ms. Stennett’s predicament.

In Los Angeles, Color Long simply had no idea how to tackle her counterfeiting problem. The 46-year-old founder of the clothing line Reignland Concept started creating, sewing and selling her children’s wear and marketing it online in 2015. Even today, she makes most of the 200 pieces of clothing she sells each month herself, with one assistant.

Last year, a friend told her that she had found her designs being sold by a store in Australia. Confused, Ms. Long contacted the proprietor, who informed her she had purchased the clothing on and had no idea the designs were originally from Reignland. Ms. Long looked at the Alibaba shop and was shocked to find a picture of her own son, one of the models in her promotional material, being used to peddle fakes of her clothing.

Ms. Long did not know she could ask Alibaba for assistance in removing these counterfeits. Instead, she contacted several shops, which, in some cases, have cooperated and removed the offending products. But Ms. Long blames the spread of copies from Alibaba for the sudden decline in sales of one of her best-selling products, a pair of children’s pants.

A few months ago, she had been selling about 30 a week, she said. Now, she moves only five a month.

Under the Table

Mr. Hankerson had worked at his father’s financial firm and in I.T. services, building websites, before he began tinkering with furniture.

He unexpectedly started Vintage in 2009, when his wife said she wanted a new patio table. Mr. Hankerson hammered one together and posted a picture on a website. He began making more tables and advertising them on the internet, too. “I loved the experience so much that I just wanted to make a better table,” he said. “Like a mad artist.”

Today, the Vintage factory is abuzz with the sounds of welding, sawing and hammering. There is one nod to modern automation — a recently acquired computerized steel-cutting machine — but much of the manufacturing is done by specially trained employees with woodworking or metalworking skills.

While his business was quickly growing, Mr. Hankerson had no idea counterfeiters were profiting off his work as well.

He first learned he had a problem in 2014, when he spotted an advertisement on Facebook that included a photo of one of his A-frame tables. He contacted the seller, who said she had bought the merchandise from elsewhere. Intrigued and worried, Mr. Hankerson began searching for images of his designs using Google, and he discovered many of them around the web.

“You would find one image, and you go and look at their catalog on Alibaba, and there’s another and another and another,” he said. “It’s going down the rabbit hole.”

Many shops promote these fakes with pictures lifted from Mr. Hankerson’s own website. One photo bouncing around Taobao features a Vintage console with framed snapshots of Mr. Hankerson and his family still hanging on the wall above it.

Some Taobao sellers admit they get their designs from Mr. Hankerson. Liao Xiaoting, a customer service provider at Shang Jie Crafts Company, said her firm’s designers use them as “reference” to make their own furniture, with changes, mainly to the materials.

Huang Geqing, proprietor of a Taobao shop called Creative Iron Factory, said that he copies Vintage’s designs, manufactures replicas in Fujian province in eastern China, and uses Mr. Hankerson’s photos to promote his own products. He first found Vintage’s designs marketed on other Taobao stores, then traced them back to the original source.

Mr. Huang estimates 100 shops may be selling fakes of Mr. Hankerson’s furniture. The style “is quite popular in China,” he said. “We can make something exactly like it.”

Mr. Hankerson said he had no idea how many of these fakes can be found outside of China. Though some of these likely counterfeits have been on Taobao, which has primarily Chinese users, he has also spotted them on the globally focused

One furniture maker in India, Raja Rani Art Handicrafts, said two pieces of furniture it sells on are based on Mr. Hankerson’s designs. “This design is from Vintage Industrial, but we also manufacture the same design in our company,” said Bobby Shukhla, who answered queries from The New York Times.

Mr. Hankerson tried contacting sellers directly, asking them to take down his images. Most of them complied. He was also encouraged when Alibaba staff members removed all of the photos he targeted on through its infringement-fighting process.

But then he ran into difficulties. Last year, the website through which he could send Alibaba requests to remove suspect listings stopped working. He tried submitting and resubmitting the required paperwork on the site, even on different computers, only to have it return error messages.

An Alibaba spokesman, in written responses, said that in October the company started an improved online system through which companies and people can report possible infringement on any of its platforms. He said that Mr. Hankerson has not tried to use this new system. Mr. Hankerson said he had but could not get it to work.

Mr. Hankerson decided to step up his efforts. In August, he sent emails to Alibaba’s investor and public relations departments threatening to raise a public ruckus about fakes on its sites.

A manager at Alibaba’s intellectual property protection department reached out. Mr. Hankerson handed over lists of more than 400 suspect images, only some of which were removed. Many of them, Mr. Hankerson was informed, were on a different Alibaba system and had to be removed through a separate process, which has not yet been undertaken.

He argued that he shouldn’t have to go to such lengths to get Alibaba to act. Once Alibaba has been made aware of his company’s intellectual property, he said, the firm should proactively prevent additional postings of his photos or clean any suspect products from its platforms. Yet despite Alibaba’s boast to the U.S.T.R., combing the millions of listings could be difficult or expensive.

Mr. Hankerson tracks images with the help of a service called Plaghunter, which both regularly monitors the use of his photos on the web, including Alibaba’s sites, and allows him to conduct a search when needed. What frustrated him the most, he said, is that the same photos keep appearing on the company’s marketplaces.

“It should be click and forget,” he said. “Our products should never be on their site.”

An Alibaba spokesman said the company had tried to work with Mr. Hankerson to remove his images and resolve his technical problems. “It’s important to note that the primary responsibility of protecting a brand rests with the brand itself,” the spokesman wrote in an email.

Additionally, Alibaba disputes the idea that it tolerates fakes. Intellectual property infringement “hurts Alibaba’s business and reputation,” the spokesman said. “It doesn’t matter whether the infringement is against big brands or small companies.”

Mr. Hankerson remains unmollified. He is talking to lawyers about his options and is trying to rally other small-business owners to the cause.

“They want to have all those counterfeits,” he said, “and they want to make money off of it.”