The Shack-Hartmann wavefront sensor is widely used to measure aberrations induced by atmospheric turbulence in adaptive optics systems. However, if strong atmospheric turbulence exists or the brightness of guide stars is low, the accuracy of wavefront measurements will be affected. In this work, we propose a compressive Shack-Hartmann wavefront sensing method. Instead of reconstructing wavefronts with slope measurements of all subapertures, our method reconstructs wavefronts with slope measurements of subapertures that have spot images with high signal-to-noise ratio. We further propose to use a deep neural network to accelerate the wavefront reconstruction speed. During the training stage of the deep neural network, we propose to add a drop-out layer to simulate the compressive sensing process, which could increase the development speed of our method. After training, the compressive Shack-Hartmann wavefront sensing method can reconstruct wavefronts at high spatial resolution with slope measurements from only a small number of subapertures. We integrate the straightforward compressive Shack-Hartmann wavefront sensing method with an image deconvolution algorithm to develop a high-order image restoration method. We use images restored by the high-order image restoration method to test the performance of our compressive Shack-Hartmann wavefront sensing method. The results show that our method can improve the accuracy of wavefront measurements and is suitable for real-time applications.