Mr. Stewart, a pioneer of satire posing as news, is coming back to a much different media landscape than the one he often skewered during his 16 years hosting “The Daily Show.” Comedy has gotten trickier with an American audience that grew more angry and divided during Mr. Stewart’s absence.

Now he’s attempting to re-engage with a show that offers fewer jokes and a more earnest agenda. With his new biweekly series, “The Problem With Jon Stewart,” his first challenge is getting people to notice it at all. Apple TV+ is decidedly more plush but less entrenched than basic cable. Comedy Central, Mr. Stewart’s old home, still has the advantage of a built-in audience for “The Daily Show,” hosted by Mr. Stewart’s successor, Trevor Noah.

Starting this week, episodes of “The Problem With Jon Stewart” will appear on Apple TV+ every other Thursday around midnight—not as a nod to late-night tradition, but because that is the time when streamers typically drop new content.

Streaming TV companies have succeeded in recruiting other talk TV giants in the second or third acts of their careers, including

Oprah Winfrey

and David Letterman, who release long-form interviews at leisurely intervals on Apple TV+ and

Netflix,

respectively. NBCUniversal bet on younger talent with “The Amber Ruffin Show,” recently renewing her for a second season on its streaming site, Peacock.

Despite the streamers’ takeover of most TV genres, from prestige dramas to trashy reality shows, they haven’t fully cracked the format of topical talk and comedy that late-night broadcast and cable shows have always owned. Netflix has tried the hardest in recent years, with a string of shows hosted by comedians, including Chelsea Handler, Joel McHale, Michelle Wolf and the late Norm Macdonald. None lasted long.

One obstacle: the way users have been trained to think about what streaming services are for, namely on-demand, anytime and binge-mode viewing. For current-affairs content, even the comedic kind, audiences tend to seek out other sources, says Geoffrey Baym, a Temple University professor of media studies. In the heyday of Mr. Stewart’s “Daily Show,” Mr. Baym published about a dozen academic papers on related topics and wrote a book called “From Cronkite to Colbert.”

“Audience habits with streaming aren’t really built around the calendar and the clock,” he says. “But satire, just by definition, is linked to current events. It’s very hard to satirize something that happened weeks or months or years ago, so you can’t satirize the headlines on streaming in the way you could on nightly television.”

One of Netflix’s more successful efforts was “Patriot Act With Hasan Minhaj,” which lasted six seasons. Each episode featured a deep dive into topics like fentanyl and global obesity by the host, a former “Daily Show” correspondent.

“Pretty much everybody else who’s doing this type of thing comes from the Jon Stewart finishing school,” says

Doug Herzog,

former president of Viacom Music and Entertainment Group, who helped launch “The Daily Show” in 1996 (with host Craig Kilborn) and oversaw it through the transition to Mr. Noah. “To a certain degree, Jon invented the genre, which probably makes it all the more imperative for him to really step into a different direction.”

Fans will find aspects of “The Problem With Jon Stewart” familiar. In front of an audience, he sits at a table for an opening monologue (now wearing a T-shirt and bomber jacket instead of a suit and tie). He twirls his pen, pauses for deadpan stares into the camera, stifles giggles behind his fist and makes self-deprecating cracks like, “I am what’s left of Jon Stewart.”

Breaking from his previous format, the show includes unscripted segments in the show’s writers’ room, where Mr. Stewart and his staff banter over each episode’s topic (expanded on in a weekly companion podcast). A separate panel discussion captures the show’s sober tone.

The show doubles down on Mr. Stewart’s style of ultrarational argument, once viewed by many of his fans as unifying and his critics as condescending. “That whole thing was: Can we talk reasonably? Can we embrace moderation? Can we work together to discuss our shared problems?” Mr. Baym says. “And unfortunately that idea seems quite quaint in our current moment.”

Mr. Stewart won’t be going head-to-head with his peers, from NBC’s Jimmy Fallon and ABC’s

Jimmy Kimmel

to Bill Maher on HBO and Greg Gutfeld on Fox News (who routinely draws more viewers than his rivals on broadcast networks). Time-slot competition and overnight Nielsen ratings don’t apply to streaming shows. Where Mr. Stewart will have to prove himself again is in the battle for attention and discussion after the fact, which can be measured in the demand for clips posted online.

In that arena, his previous show still dominates. Through last week, politically themed videos from “The Daily Show” have generated 585 million views across

Facebook,

YouTube and

Twitter

this year, according to video measurement platform Tubular Labs. In that period, which generated material on the Jan. 6 insurrection and other contentious events, “The Late Show with Stephen Colbert” came in second with 359 million views for its political clips, followed by “Jimmy Kimmel Live” (330 million).

Mr. Stewart is in the mix. His videos, including old footage and promos for his new show, accumulated 57 million views this year. That was more than the politically-themed highlights from the shows of one of his former protégés, (“Full Frontal with Samantha Bee,” 24 million), but fewer than another (“Last Week Tonight with John Oliver,” 98 million).

Write to John Jurgensen at john.jurgensen@wsj.com