Turbulence resistance is a significant research area for orbital angular momentum shift keying-based free space optical communication (OAM-SK-FSO). We put forward a two-step combinational system to receive high fidelity image data from the atmospheric turbulence (AT) channels. Firstly, the AT-detector-based multi-CNN (ATDM-CNN) demodulator is proposed which is very different from the traditional single-CNN (S-CNN) demodulator. The AT detector detects the AT strength and then an AT-determined CNN-based demodulator is activated to recognize the incident OAM modes. Sufficient numeral simulations compare the recognition rates of ATDM-CNN and S-CNN. The results indicate a tremendous improvement owing to the ATDM-CNN demodulator. Base on the ATDM-CNN's significant advantage in OAM recognition, the significant optimization of image data quality is possible in the further correction. As an option, the residual information errors are corrected by jointly using the rank-order adaptive median filter (RAMF) and the very-deep super-resolution (VDSR) network with minor information loss in severe ATs. The data increase resulting from RAMF-VDSR is tested. In conclusion, the proposed two-step system can provide a much higher quality of receiving image data in the OAM-SK-FSO link.