We denote by L(A,F) the language of words over the alphabet A avoiding the set of forbidden factors F. We provide a sufficient condition on F and |A| for the growth of L(A,F) to be boundedly supermultiplicative. That is, there exist constants C>0 and a>0, such that for all n, the number of words of length n in L(A,F) is between a^n and Ca^n. We will discuss some consequences, and an application of the same idea to circular square-free words.