In this Letter, a bidirectional amplifier configuration suppressing the relative intensity noise in a 1950-nm linearly polarized single-frequency fiber laser (SFFL) is proposed. The scheme to amplify the signal in a nonlinear saturated amplification regime with low gain distribution for suppressing the RIN is theoretically analyzed. By optimizing the input power level and reflectivity of the bidirectional power-amplifier, the RIN is decreased maximally by >24dB within the frequency range of 200 kHz. A stable output power of over 5.16 W with a polarization extinction ratio of 21.2 dB is obtained. Additionally, the amplified signal maintains a linewidth of 7.1 kHz nearly identical with that of the seed, both with a signal-to-noise ratio of more than 60 dB. This all-optical technique on noise suppression applied to the fiber amplifier paves the way to realize low-noise SFFL with power improvement.