Galleries like American Medium that work with young artists — and cater to a crowd that has grown up with the internet — are approaching web-based art with a renewed purpose.

Two of its enterprising founders, Josh Pavlacky and Travis Fitzgerald, former art students at Wesleyan, had been running an exhibition space in Portland, Ore., when Mr. Pavlacky left to join his future husband, Daniel Wallace, who was directing an art space in Philadelphia. In 2012, they contacted Mr. Fitzgerald and began thinking about how they could stake a claim in New York. American Medium created pop-up shows, staged in the Union Square loft of Mr. Fitzgerald’s father, that explored shifting boundaries between the virtual and real worlds. For the gallery’s debut, the three encouraged Jon Rafman to create a physical installation inspired by his “Brand New Paint Job” series on Tumblr, in which famous artworks were superimposed onto digital objects — Jasper Johns’s “White Flag,” for example, onto a 3-D model of the Oval Office. Mr. Pavlacky and Mr. Wallace fabricated the objects and brought them to the Union Square loft, where a Jet Ski painted in Yves Klein blue hung from the ceiling and a miniature motorcycle flaunted an Abstract Expressionist color field from Barnett Newman.

By 2014, the founders were ready to put down roots on a residential block in Bedford-Stuyvesant, Brooklyn. The three have collaborated in similar ways with other digital artists, like Brenna Murphy and Harm van den Dorpel, a group the art world has sporadically classified as Post-Internet.

“At this point we’ve staged at least over a hundred shows,” Mr. Pavlacky said, “so we know: This will look good here, this is what you do, this is a faux pas. And that information is really helpful for someone who only makes collages on the computer.”

Even when the digital component isn’t as obvious, the work is often still influenced by the internet. In “Lavendra,” its current show, the artist E. Jane extracted images of ’90s R&B singers from YouTube videos and printed them on fabric. E. Jane says the resolutions were lower than those found on old videos featuring white pop singers because not many people cared to upload higher-quality versions; and this reflects the limited staying power of black female artists in popular culture. Those who are internet savvy will recognize in the show a cutesy aesthetic that has proliferated online, called cybertwee.

Soon, American Medium will be relocating to Chelsea, steps from the High Line and down the block from David Zwirner. The founders think their artists deserve to be shown there and, as Daniel Wickerham, one of those artists, also speculated when discussing the move, “They want to win.”