A Generative Adversarial Network (GAN) with generator G trained to model the prior of images has been shown to perform better than sparsity-based regularizers in ill-posed inverse problems. Here, we propose a new method of deploying a GAN-based prior to solve linear inverse problems using projected gradient descent (PGD). Our method learns a network-based projector for use in the PGD algorithm, eliminating expensive computation of the Jacobian of G. Experiments show that our approach provides a speed-up of 60-80x over earlier GAN-based recovery methods along with better accuracy in compressed sensing. Our main theoretical result is that if the measurement matrix is moderately conditioned on the manifold range(G) and the projector is delta-approximate, then the algorithm is guaranteed to reach O(delta) reconstruction error in O(log(1/delta)) steps in the low noise regime. Additionally, we propose a fast method to design such measurement matrices for a given G. Extensive experiments demonstrate the efficacy of this method by requiring 5-10x fewer measurements than random Gaussian measurement matrices for comparable recovery performance. Because the learning of the GAN and projector is decoupled from the measurement operator, our GAN-based projector and recovery algorithm are applicable without retraining to all linear inverse problems in which the measurement operator is moderately conditioned for range(G), as confirmed by experiments on compressed sensing, super-resolution, and inpainting.