Atmospheric turbulence (AT) will cause the crosstalk between OAM (orbital angular momentum) beams, which makes it difficult to identify the OAM modes at the demodulation part. In order to reduce AT's interference, we propose to combine phase compression (PC) with an improved convolutional neural network (CNN) to achieve high-precision recognition of OAM modes. We have investigated the performances (OAM recognition accuracy) of our system with changes of AT's intensities, transmission distances, and PC ratios. The results demonstrate that compared with traditional recognition systems, our model still has higher accuracy in the case of strong AT and long-distance transmission, which can be attributed as the additional characteristics of the PC OAM beams. At the same time, we also investigate the performances of the multiplexed hybrid vortex beam under different AT intensities, and discuss the recognition of the multiplexed hybrid vortex beam with the same OAM mode but different PC ratios, where the results show that the multiplexed hybrid vortex beams with the same OAM mode but different PC ratios can also be recognized at the receiving part. In addition, we also used unknown turbulence to test our trained model, and our model demonstrates good generalization ability, which has superior robustness for the unknown AT environments.