Generative adversarial networks (GANs) train implicit generative models through solving minimax problems. Such minimax problems are known as nonconvex- nonconcave, for which the dynamics of first-order methods are not well understood. In this paper, we consider GANs in the type of the integral probability metrics (IPMs) with the generator represented by an overparametrized neural network. When the discriminator is solved to approximate optimality in each iteration, we prove that stochastic gradient descent on a regularized IPM objective converges globally to a stationary point with a sublinear rate. Moreover, we prove that when the width of the generator network is sufficiently large and the discriminator function class has enough discriminative ability, the obtained stationary point corresponds to a generator that yields a distribution that is close to the distribution of the observed data in terms of the total variation. 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We establish global convergence to optimality for IPM-based GANs where the generator is an overparametrized neural network. 