In next-generation wireless communications systems, accurate sparse channel estimation (SCE) is required for coherent detection. This paper studies SCE in terms of adaptive filtering theory, which is often termed as adaptive channel estimation (ACE). Theoretically, estimation accuracy could be improved by either exploiting sparsity or adopting suitable error criterion. It motivates us to develop effective adaptive sparse channel estimation (ASCE) methods to improve estimation performance. In our previous research, two ASCE methods have been proposed by combining forth-order error criterion based normalized least mean fourth (NLMF) and L1-norm penalized functions, i.e., zero-attracting NLMF (ZA-NLMF) algorithm and reweighted ZA-NLMF (RZA-NLMF) algorithm. Motivated by compressive sensing theory, an improved ASCE method is proposed by using reweighted L1-norm NLMF (RL1-NLMF) algorithm where RL1 can exploit more sparsity information than ZA and RZA. Specifically, we construct the cost function of RL1-NLMF and hereafter derive its update equation. In addition, intuitive figure is also given to verify that RL1 is more efficient than conventional two sparsity constraints. Finally, simulation results are provided to confirm this study.