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Advancements in optical communications have increasingly focused on leveraging spatial-structured beams such as orbital angular momentum (OAM) beams for high-capacity data transmission. Conventional electronic convolutional neural networks exhibit constraints in efficiently demultiplexing OAM signals. Here, we introduce a hybrid optical-electronic convolutional neural network that is capable of completing Fourier optics convolution and realizing intensity-recognition-based demultiplexing of multiplexed OAM beams under variable simulated atmospheric turbulent conditions. The core part of our demultiplexing system includes a 4F optics system employing a Fourier optics convolution layer. This optical spatial-filtering-based convolutional neural network is utilized to realize the training and demultiplexing of the 4-bit OAM-coded signals under simulated atmospheric turbulent conditions. The current system shows a demultiplexing accuracy of 72.84% under strong turbulence scenarios with 3.2 times faster training time than all electronic convolutional neural networks. Optical beams carrying orbital angular momentum (OAM) are promising candidates for free-space optical communication. The authors devise a hybrid optical-electronic convolutional neural network approach reaching a 4-bit OAM-coded signal demultiplexing accuracy of 72.84% under strong atmospheric turbulence conditions with 3.2 times faster training time than all electronic convolutional neural network. [ABSTRACT FROM AUTHOR]