Objective: Synthetic aperture sonar (SAS) achieves high-resolution underwater imaging by synthesizing a large virtual aperture through platform motion. However, system parameters still constrain SAS imaging resolution, especially under low-frequency detection conditions. Additionally, echoes from small underwater targets are usually weak and can be easily contaminated by background noise and interference, which further reduces imaging quality and target detectability. To improve imaging resolution and enhance weak target echoes, this study proposes a weighted regularization-based signal enhancement SAS method, termed RSE-SAS. This method exploits the sparse spatial distribution characteristics of small underwater targets and aims to improve target focusing while suppressing background noise. Methods: A multisubarray SAS signal model was first established, and the imaging region was discretized into candidate scattering grids. Rather than directly fusing the signals collected by multiple subarrays, a mapping space for the imaging region was constructed using theoretical multichannel echoes from different candidate scattering positions. The received echoes were then mapped into this space. As a result, the imaging problem was transformed into the estimation of sparse projection coefficients. Based on the sparse distribution prior of small targets, an L1-norm regularization term was introduced to formulate the imaging problem as a sparse reconstruction problem. An iterative shrinkage-thresholding algorithm based on the majorization–minimization framework was employed for optimization. To alleviate amplitude underestimation caused by the fixed threshold in conventional sparse reconstruction, a weighted regularization strategy was further introduced. Target-related coefficients with relatively large amplitudes were assigned smaller penalty thresholds. In contrast, noise- and interference-related coefficients were subjected to stronger penalties. Therefore, target echoes were preserved and enhanced while background components were gradually suppressed during iteration. Point- and extended-target simulations were conducted. DAS, RD, conventional CS-SAS, and the proposed RSE-SAS were then compared in terms of imaging resolution, target detection probability, output signal-to-noise ratio, convergence performance, and computational efficiency. Results: The simulation results showed that RSE-SAS achieved better target focusing and background suppression than the comparison methods. For point-target imaging, DAS and RD exhibited relatively broad mainlobes, whereas CS-SAS improved target focusing while retaining noticeable background noise. RSE-SAS further suppressed the background components while maintaining a concentrated target response. Similar improvements were observed for extended-target imaging. Quantitative analysis based on 100 Monte Carlo trials showed that the azimuth and range −3 dB mainlobe widths of RSE-SAS were 0.29 and 0.81 m, respectively, compared with 1.46 and 1.82 m for DAS. The target detection probability increases from 0.77 for DAS to 0.98 for RSE-SAS, while the output signal-to-noise ratio increased from 12.3 to 31.6 dB. Compared with CS-SAS, RSE-SAS reduced the azimuth and range mainlobe widths by approximately 35.2% and 12.9%, respectively. It also increased the detection probability by 0.05 and improved the output signal-to-noise ratio by 3.1 dB. Furthermore, the target-related coefficients stabilized after approximately 10 iterations. With the same data scale and computational platform, the average single-frame imaging time for RSE-SAS was 52 ms, indicating that weighted regularization incurred limited additional computational cost. Conclusions: The proposed RSE-SAS method combines the sparse spatial prior of small underwater targets with a weighted regularization strategy. By applying different penalty strengths to target- and noise-related coefficients, this method alleviates amplitude underestimation in conventional sparse reconstruction and realizes simultaneous target echo enhancement and background suppression. The results demonstrate that RSE-SAS provides improved imaging resolution, target detection performance, noise robustness, and focusing capability while maintaining acceptable convergence and computational efficiency. The proposed method therefore has potential for high-resolution imaging of small underwater targets. Future work will focus on validation using measured SAS data and further evaluation under more complex underwater propagation conditions.