In this paper we analyze the numerical behavior of first-order quadratic penalty methods for solving large-scale conic constrained convex problems with composite objective function. Contrary to the most of the results on penalty methods, in this work we do not assume the existence of a finite optimal Lagrange multiplier. We derive the iteration complexity of the classical quadratic penalty method, where the corresponding penalty regularized formulation of the original problem is solved using Nesterov's fast gradient algorithm. We provide rate of convergence results in terms of feasibility violation and suboptimality for adaptive and non-adaptive variants of the penalty scheme, under various assumptions on the composite objective function. Finally, we show on a simple example that our complexity estimates are tight.