Belief propagation (BP) decoding has been recognized for its capacity-approaching performance and high throughput when decoding long low-density parity-check (LDPC) codes. However, the application of BP decoding for short codes is hindered by dense parity-check matrices (PCMs) and prevalent short cycles in the Tanner graph. In this paper, we introduce a general method to extract an optimized sparse PCM for short binary block codes, which removes length-four cycles and enhances the connectivity of short cycles to enable BP decoding with improved performance. Notably, for short binary codes with lengths up to 64, our BP decoding performance approaches the maximum likelihood bound and surpasses the best-reported BP results with reduced computational complexity. Compared with other universal decoding algorithms, BP decoding using our extracted sparse PCMs is competitive in terms of both error-rate performance and computational complexity. These promising results suggest that our method to improve BP decoding for short codes is a step toward a practical universal BP decoder for next-generation communication systems.