Reconfigurable intelligent surface (RIS) can improve the coverage range and signal quality of wireless networks. But its performance depends on the estimation accuracy of channel state information (CSI). The compressed sensing based on stage-wise orthogonal matching pursuit (StOMP) algorithm can be employed to estimate the CSI, however, the hard threshold methods involved in the traditional StOMP can introduce the undesired noise components and redundant atoms, thereby reducing the accuracy of estimated CSI and even lower the convergence speed. To this end, a novel channel estimation approach using the improved StOMP algorithm is proposed in RIS-assisted system to improve the channel estimation quality while achieving the high-speed convergence. By utilizing the difference between residual ratios of adjacent iterations, a normalization procedure can be performed to optimize the iterative stopping conditions, and the algorithm can effectively reduce the impact of out-of-band noise while optimizing the iteration stopping condition with normalized residual ratios. Furthermore, it can reduce redundant atom selection and leads to faster system operation. The results demonstrate that the proposed method can reduce the run time by 5 times at a relatively low complexity with favorable estimation accuracy. Moreover, it can also achieve 86% reduction in the selection of redundant atom.