My early adulthood was, even during this era, called the Golden Age of Television. It was a period in which the medium ascended in quality (and budget) and there was a lot more serious consideration of TV shows as an art from to rival cinema. Indeed, as movies began playing it safer, with more and more franchise-focused projects (I will defend the MCU as being an original experiment in blockbuster movie-making that everyone just tried to copy, though even I have gotten kind of tired of it in the post-Endgame stretch these past 5 years,) we got TV shows that were becoming the place for talented actors to showcase their skills, allowing for gradual and thorough character development over time.
While the shows that were seen as "high art" usually starred villainous protagonists and antiheroes like The Sopranos, Mad Men, and Breaking Bad, the era also saw more ambitious genre shows. Lost, for example, was the big Water Cooler show, with Game of Thrones probably taking that spot over shortly after Lost ended (or did they overlap?)
Anyway, the format of television shows for much of their lifespan was episodic: you'd have a premise with a regular cast of characters, and each episode (each week) they'd be dealing with some new challenge or conflict. Many shows were structured in such a way to encourage this kind of storytelling. Police procedurals, medical procedurals, and fantasy/sci-fi "monster of the week" shows are all built around having some kind of mystery to solve. In most cases, the mystery is resolved by the end of the episode, and the characters reset to their starting positions, essentially, in a happy equilibrium, before the next episode challenges them.
Now, on a practical level, this style of storytelling could be very helpful when the show went into syndication: if the plot of each episode is self-contained, then you don't need to watch the episodes in order. You could hop into a season four episode of Star Trek the Next Generation, and then watch a season six episode next, and you should be able to follow both relatively easily.
The Simpsons, hardly a procedural, at least used to make something of a joke about this zero continuity structure: in one infamous episode, milquetoast elementary school principle Seymour Skinner is revealed to actually be an imposter who stole the identity of his sergeant back in Vietnam after the real Skinner supposedly died. However, when the real Skinner is finally released from a Vietnamese P.O.W. camp, the false Skinner, actually named Armin Tanzarian, is ousted from his job and the community struggles to adjust. But, by the end of the episode, basically just because the town feels weird about this change, a judge rules that Tanzarian will now be considered the real Skinner, that the actual Skinner is banished from the town, and that no one shall mention it again under penalty of torture.
Still, even in the 1990s, the idea of recognizing some continuity over the course of a series wasn't entirely out of the question. Star Trek TNG had, for example, the ongoing conflict of Worf with the Duras clan, as well as Picard's struggles to reckon with his PTSD after being briefly assimilated by the Borg. Notably, Deep Space Nine, the Star Trek show that slightly overlapped with TNG, eventually put a lot of its focus on the overall arc of the Dominion War, an arc that took, if memory serves, over half the series' length, starting in full in season four.
Buffy the Vampire Slayer, which straddled the turn of the millennium, had an interesting approach: most episodes told a self-contained story, but each season had its Big Bad who had to be dealt with in the season finale. While the villainous threats didn't tend to last more than one season (except Spike, who gradually evolves from a villain to an ally) the ongoing weight of the events that transpire does take its toll on the characters.
But I think it was in the late 2000s/early 2010s that we started to see shows really go full-serialized. Some shows probably should have done so earlier: I remember watching Dollhouse, Joss Whedon's post-Firefly TV project before he directed the Avengers and before the world discovered that he's apparently a piece of shit, and feeling infuriated that the show was so slow for any long-term plot to finally take effect.
So, it was around this time period that Fringe came out. Premiering four years after Lost did, it was J.J. Abram's next big thing (while I have no reason to think that Abrams is a bad human being, I have grown very skeptical of him as an artist, starting around Star Trek Into Darkness, as I feel that he largely just re-creates the movies that he loved from the 1980s without contributing any real new ideas). Like Lost, though, I think Abrams' involvement in the show was only at the outset.
To start, the show was very much in the vein of The X-Files - a group of investigators who looked into paranormal events, though while the X-Files lived more in the land of cryptids, UFOs, and conspiracy theories, Fringe focused more on "fringe science," the kind of weird experimentation that went on in secret during the Cold War, this time under the storm cloud that was the War on Terror. There's a lot of victims in this show who are randomly selected but deliberately targeted, much as terrorists will pick unlucky civilians to die in their attacks.
The show starts off with (though it's a throughline) a lot of Body Horror, but to begin with, most episodes are your classic case-of-the-week structures. Yes, there's an ongoing plot in which central protagonist Olivia Dunham has a lingering phantom of her dead lover's memories in her mind, the after-effect of a mad experiment intended to help her recover some of his memories while he was comatose and dying from a strange synthetic toxin.
Season one does introduce David Robert Jones and the ZFT terrorist organization as primary threats, but it only touches on these guys in some episdoes.
Where things really transition, though, is about halfway or two thirds of the way through season two. After a story hinted at but never fully confirmed gets its full episode-long flashback, the plot begins to laser-focus on what will be pretty central to the whole series, which is the alternate universe.
Spoilers from here on, I guess?