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Relativistic magnetized shocks are natural sources of coherent radiation, representing a promising framework for fast radio bursts (FRBs). This Letter explores how the radiation reaction (RR) effect, triggered by high-energy photon emissions during shock radiation, significantly alters particle dynamics and coherent radiation properties. Using kinetic particle simulations, we demonstrate that RR severely suppresses electron energies from shock acceleration, resulting in multiple coherent gyration cycles at the shock front. It amplifies the intensity of coherent radiation and boosts energy efficiency by several fold, as compared to the single gyration cycle in the standard model of relativistic magnetized shocks. We further find that the coherent radiation spectrum from RR-mediated shocks is characterized by upshift peak frequency, broaden bandwidth, and narrow spectral peak. These RR-induced radiation changes may be related to several observed FRB phenomena, including the statistically positive correlation between luminosity and bandwidth in repeating and one-off FRBs, the narrow spectra seen in some FRB events, and the bimodal energy distribution reported in FRB 20121102A.