My understanding of renormalizability is that a theory is renormalizable if it the divergences in its amplitudes can be cancelled out by finitely many terms. I see that by adding counterterm (in the MS-bar scheme)
the one-loop divergence of QED can be made finite. However, I do not see how this makes QED renormalizable? Surely as we work with diagrams with more loops, we will get more counterterms - given that we can have diagrams with arbitrarily many loops, do we not need an infinite number of counterterms to cancel these out?