A novel joint atmospheric turbulence (AT) detection and adaptive demodulation technique based on convolutional neural network (CNN) are proposed for the OAM-based free-space optical (FSO) communication. The AT detecting accuracy (ATDA) and the adaptive demodulating accuracy (ADA) of the 4-OAM, 8-OAM, 16-OAM FSO communication systems over computer-simulated 1000-m turbulent channels with 4, 6, 10 kinds of classic ATs are investigated, respectively. Compared to previous approaches using the self-organizing mapping (SOM), deep neural network (DNN) and other CNNs, the proposed CNN achieves the highest ATDA and ADA due to the advanced multi-layer representation learning without feature extractors designed carefully by numerous experts. For the AT detection, the ATDA of CNN is near 95.2% for 6 kinds of typical ATs, in cases of both weak and strong ATs. For the adaptive demodulation of optical vortices (OV) carrying OAM modes, the ADA of CNN is about 99.8% for the 8-OAM system over the computer-simulated 1000-m free-space strong turbulent link. In addition, the effects of image resolution, iteration number, activation functions and the structure of the CNN are also studied comprehensively. The proposed technique has the potential to be embedded in charge-coupled device (CCD) cameras deployed at the receiver to improve the reliability and flexibility for the OAM-FSO communication.