The Shack-Hartmann wavefront sensor (SH-WFS) is known to produce incorrect measurements of the wavefront gradient in the presence of non-uniform illumination. Moreover, the most common least-squares phase reconstructors cannot accurately reconstruct the wavefront in the presence of branch points. We therefore developed the intensity/slopes network (ISNet), a deep convolutional-neural-network-based reconstructor that uses both the wavefront gradient information and the intensity of the SH-WFS's subapertures to provide better wavefront reconstruction. We trained the network on simulated data with multiple levels of turbulence and compared the performance of our reconstructor to several other reconstruction techniques. ISNet produced the lowest wavefront error of the reconstructors we evaluated and operated at a speed suitable for real-time applications, enabling the use of the SH-WFS in stronger turbulence than was previously possible.