In this paper, a novel turbo-coded 16-ary orbital angular momentum - shift keying-free space optical (OAM-SK-FSO) communication system combining a convolutional neural network (CNN) based adaptive demodulator under strong atmospheric turbulence is proposed for the first time. The feasibility of the scheme is verified by transmitting a 256-grayscale two-dimensional digital image. The bit error ratio (BER) performance of the system is investigated and the effect of different factors such as turbulence strength, propagation distance, code rate, length of random interleaver and length of bit interleaver is also taken into account. An advanced encoder/decoder structure and mapping scheme are applied to diminish the influence of CNN misclassification and reduce the BER effectively. With the optimal encoder/decoder structure and CNN model settings, the BER varies from 0 to 4.89×10-4 when the propagation distance increases from 200m to 1000m for a given turbulence strength Cn2 equals 5×10-14m-2/3. For a determined propagation distance equals 400m, the BER ranges from 0 to 4.01×10-4 when Cn2increases from 1×10-15m-2/3 to 4×10-13m-2/3. Our numerical simulations demonstrate that the proposed system can provide better BER performance under strong atmospheric turbulence and conditions when the classification ability of CNN is limited.