It is a known fact that if a group $G$ has finitely many commutators then the order of the commutator subgroup is finite. I have found the following upper bound: Let the number of commutators be equal to $n<\infty$ then $$|G'|< n^{2n^2\log_2(n)}$$ I did this by combining the best bound given by a proof of Schur's theorem and the best bound given by a proof of the reverse Schur's theorem for finitely generated groups.
This upper bound does not seem very good. Are there better known bounds? For example by using a different method.