In AmE, I hear "rooting for" far more often than "support" in this sense, and I pretty much never hear "supporters" used to mean "fans of". But "Which team are you a fan of" is more common yet. From my reading, "support" and "supporters" are indeed common in BrE.
Oddly, "support" is used quite often in AmE is similar but non-sports contexts:
- I support the Bach Consort (usually meaning that I donate money)
- I support the Purple party (meaning that I vote for them, or advocate for their policies)
- I support gender equality. (meaning that I am in favor of it)
I don't know why US-English doesn't commonly use "support" ,in a sports context, but it doesn't seem to do so very often.