We consider fully convex lower level standard optimistic bilevel problems. We show that this class of mathematical programs is sufficiently broad to encompass significant real-world applications. We establish that the critical points of a relaxation of the original problem correspond to the equilibria of a suitably defined generalized Nash equilibrium problem. The latter game is proven to be convex and with a nonempty solution set. Finally, leveraging this correspondence, we provide a provably convergent, easily implementable scheme to calculate critical points of the relaxed bilevel program. As witnessed by some numerical experiments on an application in economics, this algorithm turns out to be numerically viable also for big dimensional problems.