Turbo codes have been successfully adopted in 4G LTE, which can approach the channel capacity with Bahl-Cocke-Jelinek-Raviv (BCJR) decoding. With the evolution from 4G LTE to 5G NR, there is a demand to design a unified channel decoder that supports both LTE Turbo codes and NR low-density parity-check (LDPC) codes. One solution is to employ belief propagation (BP) decoding on the bipartite Tanner graph for both codes. However, although MacKay pointed out that Turbo codes have a sparse parity-check matrix, the existence of 4-cycles in such a matrix severely deteriorates the performance of BP decoding. In this paper, we propose two polynomial-based methods to optimize the parity-check matrix of Turbo codes by improving the sparsity while also removing 4-cycles and even 6-cycles compared to the original matrix. Simulation results show that the improved BP decoding for Turbo codes halves the error-correction performance gap between the original BP decoding and BCJR decoding, which is a promising step towards the unified channel decoder design based on the BP algorithm.