The traditional belief-propagation (BP) decoder for low-density parity-check (LDPC) codes assumes that the soft input information for each bit is independent of all other bits. However, for partial-response (PR) channels, the log-likelihood ratios (LLRs) generated by the channel detector are correlated. In this paper, we propose an improved BP (IBP) decoder to take into account the dependence among the LLRs produced by channel detectors. The IBP decoder needs not only the LLR for each bit but also the probability distribution of the subblock that contains this bit. Therefore, we revise the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm accordingly to generate the additional information during channel detection. We apply the proposed decoder to an ideal PR channel and, as a practical example, to equalized perpendicular magnetic recording channels (PMRCs). The IBP decoder significantly outperforms the traditional BP decoder without turbo equalization, but this gain is substantially reduced when we introduce turbo equalization. Furthermore, we extend the IBP decoder to nonbinary LDPC codes, and show that we can achieve additional gains.