If you're being that pedantic, C itself is not Turing-complete in the first place: pointers have finite size, which means that C can address only a bounded amount of memory, so it is a finite state machine (just with an extremely large number of states).
I don't think it's about that. Technically, everything is finite. But placing limits on the loops and restricting memory allocation bounds the complexity. It probably limits everything to P-time.
Without dynamic memory allocation? I think you'd need at least O(n) memory to store enough state to go exponential. But it's some time I did complexity theory; I could be wrong.
