^Technically the original Superman series is 5 films... Superman returns is better than the fourth though.
Superman Returns seems to forget the events of III and IV for some reason.
The reason being that they have been removed from the series by the studio, after the fact. When Richard Donner was allowed back to recut Superman II, the studio decided to remove three superman films from the canon to preserve continuity the best they could: Superman II (Lester Cut), Superman III, and Superman IV: A Quest for Peace have all been removed.
It now runs Superman : The Movie, Superman II (Donner Cut), Superman Returns.
It has to be this way for the kid in Returns to exist at all. As a fan of the originals growing up, I started noticing the issues with Lester's cut after not seeing it for a long time, and then trying to give it a watch sometime in the early 90s. The film is GOOFY, has way too many sight gags and silly one-liners, and for the love of Pete, Superman makes a SOUFFLE with his heat vision...That was just...unacceptable.
Although it is clearly still flawed, the Donner cut is MUCH better, and ties the events of the first film in with the villains arrival on Earth is part 2, which was the original intention before Donner got fired, Lester took over, and Brando demanded ridiculous amount of money to be in the film, causing him to be removed from the proceedings entirely, eventually being replaced by Lara El.
And that's it. Man of Steel is NOT connected to the films in any way.