My combat ships are getting xp and leveling up, but only after winning battles.

So the original poster should be seeing some leveling, where he says he's seeing none.

Anyway, do you know how it works?  If, for example, it took more and more experience for each level-up, it might appear to stop if you're using the same fleet over and over.  If it's a fixed number of XP per level then it shouldn't seem to stop.

I noticed my carriers are not leveling, could that be because they use fighter?

I didn't really pay attention to my earlier combat vessels, but will test them out again.

Edit: Tested it out and I was mistaken. Leveling seems fine