As far as I understand it, in the Wilsonian picture of renormalization, we view a theory as having some fixed cutoff and bare couplings, and integrate out high-momentum modes to understand what happens at low momentum. We say that an operator is relevant if its coupling constant grows when we go to low momentum scales, and irrelevant if it shrinks.
Now, in the "usual" picture of renormalization, we have a QFT which we want to define as a continuum limit of a theory with a cutoff, i.e. the limit where the cutoff goes to infinity. We want to take this limit holding the physical coupling constants at some energy scale to be fixed; to do this, we add cutoff-dependent counterterms to the bare Lagrangian. We say that an interaction is renormalizable if we only need to add a finite number of counterterms, and non-renormalizable if we need an infinite number of counterterms.
However, I don't understand how these two pictures fit together. In particular, it is usually stated that irrelevant operators are non-renormalizable, and relevant operators are renormalizable, but this doesn't seem obvious to me. Can someone explain why this is true?