The conventional belief propagation (BP) of the low-density parity-check (LDPC) is designed based on additive white Gaussian noise (AWGN) close to the Shannon limit; however, the correlated noise due to chromatic dispersion or square-law detection results in a performance penalty in the intensity modulation and direct-detection (IM/DD) system. We propose an iterative BP cascaded convolution neural network (CNN) decoder to mitigate the correlated channel noise. We use a model of correlated Gaussian noise to verify that the noise correlation can be identified by the CNN and the decoding performance is improved by the iterative processing. We successfully demonstrate the proposed method in a 50-Gb/s 4-ary pulse amplitude modulation (PAM-4) IM/DD system. The simulation results show that the proposed decoder can achieve a BER performance improvement which is robust to transmission distance and launch optical power. The experimental results show that the iterative BP-CNN decoder outperforms the standard BP decoder by 1.2 dB in received optical power over 25-km SSMF.