Accurate channel estimation is essential for broadband wireless communications. Adaptive sparse channel estimation schemes based on normalized least mean square (NLMS) have been proposed to exploit channel sparsity for improved performance. However, their performance bound as derived in this paper indicates that the invariable step size (ISS) usually used for iteration in these schemes would lead to performance loss or/and slow convergence speed as well as high computational cost. To solve this problem, based on the observation that a large step size is preferred for fast convergence while a small step size is preferred for accurate estimation, we then propose to replace the ISS by the variable step size (VSS) to improve the performance of sparse channel estimation. The key idea is that the VSS can be adaptive to the estimation error in each iteration, i.e., a large step size is used in the case of large estimation error to accelerate the convergence speed, while a small step size is used when the estimation error is small to improve the steady-state estimation accuracy. Finally, simulation results verify that better mean square error (MSE) and bit error rate (BER) performance could be achieved by the proposed scheme.