5G new radio low density parity check (5G-NR LDPC) codes specified by the 3rd generation partnership project (3GPP), a representative class of multi-edge-type, quasi-cyclic and raptor-like LDPC codes with rate-compatible and length-scalable capability, could meet with high-throughput demands. The corresponding decoding algorithm and implementation should support low complexity, high performance and scaleinvariance to channel variations and receiver imperfections. Flooding scheduling is usually used for the prevalent decoding algorithms and architecture of LDPC codes, but it is inefficient and inflexible compared with layered scheduling. This paper proposes a new simplification method where the sub-groups with similar fitting functions of normalization factors are incorporated into the same type, and enhanced adaptive normalized min-sum algorithm (enhanced ANMSA) for layered scheduling is further proposed for the implementation of 5G-NR LDPC codes. Based on the new simplification method, the proposed algorithm defines three types of check nodes for base graph 2 (BG2) in 5G-NR LDPC codes, and then obtains three look-up tables for BG2 by training. Simulation results demonstrate that the proposed algorithm can achieve excellent performance for 5G-NR LDPC codes of different rates and lengths over Binary-input AWGN channel and Rayleigh channel.