The film and television industry perhaps more than ever works on existing franchises rather than new original content. Sequels and spin-offs are extremely common, but the prequel is starting to build traction again. There are now plenty of TV series which act as predecessors to huge movies or well-known shows.
Sometimes allowing creative freedoms within the writing team though means that some continuity issues can occur, between the prequel and the film or show it's leading in to. It might also be based on the casting, a characteristic that has been focused on, or perhaps the narrative overall doesn't make sense in the much larger context.
Gotham (2014-2019)
DC's Gotham wasn't necessarily a direct prequel to any film or TV show, however, it was supposed to be an origin story for Batman in general, tracking the life of a young Bruce Wayne growing up in the titular city. Unfortunately, because of the premise, a number of things ended up not making a lot of sense.
Sure there was a lot of evolution in the show between the first episode and the final season, but some of the villains whose origins are so deeply tied to the dark knight in the comics had vastly different stories instead. What's more, some of the continuity became confused with multiple versions of the Joker being created; a strange combination of Jerome and his twin, which doesn't add up long term.
Better Call Saul (2015-)
Better Call Saul had a lot to live up to. It had to introduce a different side of Saul Goodman, considering fans were already so familiar with the character. However, it also needed to still exist within the Breaking Bad universe.
The result of this is a number of continuity issues and larger questions. The number of wives he has for instance, as mentioned in Breaking Bad doesn't really line up with the prequel and sometimes storylines have to be dropped entirely.
The Mandalorian (2019-)
The The Mandalorian that has yet to be addressed though.
In the season 2 finale, Luke Skywalker arrives to take Grogu away. For such a powerful creature to be trained under Luke though, there should be an appearance at the academy in the comic spin-offs or a mention in the newest trilogy of movies. Considering the importance of the creature this is a major flaw in the continuity of the overall story; perhaps season 3 will provide some answers though.
Terminator: The Sarah Connor Chronicles (2008-2009)
In theory, Terminator 2 and a prequel to Terminator 3 and the following films. The continuity of all of these projects is extremely flawed though and it's difficult to understand what's part of the canon universe.
In many ways, the show actually disregards any movies which come after its release, despite the fact that it should be a prequel explaining them. Even though the latest Dark Fate tried to fix the timeline, there are even more glaringly obvious continuity issues thanks to the series. Perhaps it's no longer even in canon anymore; it's hard to tell with the reboots of the universe.
Young Sheldon (2017-)
The Big Bang Theory. While there are minor plot points that don't completely add up between the two shows, there's one huge continuity issue that audiences have pointed out.
It's actually a casting problem, with the actor who plays Sheldon's father also appearing as a bully in The Big Bang Theory. Considering it's a memorable episode, viewers have easily ed the connection, but it's really something that will just have to be ignored.
Agent Carter (2015-2016)
There's a big debate as to whether TV shows like Avengers: Endgame.
However, if Agent Carter is part of the canon there's a lot of explaining to do in regards to how S.H.I.E.L.D. was actually set up. The events in the TV show don't correspond with how the rest of the MCU claims that Carter created the agency. What's more, this could get more complicated Hawkeye.
Rock And Chips (2010-2011)
Adding a prequel to an ongoing sitcom like the British classic Only Fools And Horses is likely only going to confuse the continuity. It doesn't matter as much on shows like these, as everything is treated in a bit of a tongue-in-cheek sort of way.
However, Rock And Chips the prequel show did include a number of errors that don't fully match up to the show it's leading into. From the ages of the characters in each series to the years certain people died and even some simple personality traits, details are forgotten about in favor of new jokes.
The Young Indiana Jones Chronicles (1992-1993)
Indiana Jones canon with Harrison Ford even appearing as the titular character in a few of the episodes. That doesn't mean that the continuity is always protected though.
Part of the issue usually stems from the age of the character and how it can be possible that he is actually part of certain events at specific times. From appearances during historical moments to lessons learned that are later ignored in the film series; the alien story of The Crystal Skull perhaps is the movie that disregards the series the most.
Fear The Walking Dead (2015-)
The Walking Dead, which aims to try and demonstrate how the zombie population was formed in the first place. It has been criticized for its continuity issues, even within the show itself; let alone connections to the main show.
Narratively speaking though, the zombies themselves have to get more intelligent to increase the conflict and the outbreak has to develop in an interesting way. The way that these elements are actually portrayed though doesn't feel as if they line up with some components in The Walking Dead, making it insignificant as a prequel.
Star Wars: The Bad Batch (2021-)
Acting as a prequel to the next saga of the story, The Bad Batch tried to show the events which got Kanan's master killed. However, this flies in the face of the continuity of both Rebels and the spin-off comics which Bad Batch should also be acting as a prequel to. It's a major canonical error that is hard to ignore.