Cowboy Bebop (ESPECIALLY the dub) was very much written and "acted" in a "western" style. Same with contemporaries like Outlaw Star and Big O (which is very much a referene to Batman).

In general? Rowdy Roddy Piper once said (paraphrasing) that pro wrestling is about explosive acting and emoting and film is about implosive acting and emoting. And ask anyone who has done theatre (which pro wrestling basically is)) and they'll say something similar. Because when the camera zooms in to that single tear in Ryan Gosling's eye? We can see it. When Ryan Gosling is on the stage behind twelve layers of plexiglass so we don't try to lick his sweat? We can't see that tear and he needs to make a big gesture to make it clear he is having an emotional breakdown.

And that applies to animated content. With very few exceptions (almost all short films), you can't have the level of detail where such a subtle response will be noticed. Even the truly amazing finales of Cowboy Bebop was built on us having learned enough about the characters to fill in the gaps by that point.

So you get the very over exaggerated "acting" styles so that you know this fairly hastily drawn person is really feeling it. And that is true regardless of the country of origin for the cartoon.