I'm not sure about other Americans, but I'll sometimes say "swim[ming] trunks" if I'm specifically referring to male swimming attire. Otherwise, yeah, it's often "swim[ming] suit" or "bathing suit" here.
Save for one singular design which, it would seem, distinguishes itself with such remarkable uniqueness as to warrant the bestowal of its very own appellation: «budgie smugglers» (AusE).
As a kid I found the Irish usage of “press” for cupboard and “runners” for trainers/sneakers confusing as hell.