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Atmospheric effects, such as turbulence and background thermal noise, inhibit the propagation of coherent light used in ON-OFF keying free-space optical communication. Here we present and experimentally validate a convolutional neural network to reduce the bit error rate of free-space optical communication in post-processing that is significantly simpler and cheaper than existing solutions based on advanced optics. Our approach consists of two neural networks, the first determining the presence of coherent bit sequences in thermal noise and turbulence and the second demodulating the coherent bit sequences. All data used for training and testing our network is obtained experimentally by generating ON-OFF keying bit streams of coherent light, combining these with thermal light, and passing the resultant light through a turbulent water tank which we have verified mimics turbulence in the air to a high degree of accuracy. Our convolutional neural network improves detection accuracy over threshold classification schemes and has the capability to be integrated with current demodulation and error correction schemes.