The short-term sensitivity of a several-kilometers long fiber-optic gyroscope is limited mainly by thermal phase noise and relative intensity noise. Increasing the phase modulation frequency decreases the thermal phase noise but not the relative intensity noise since it behaves as white noise. Here, we propose and experimentally demonstrate that the angular random walk can be effectively decreased by suppressing relative intensity noise at the modulation frequency and its third-order harmonic using direct feedback to the drive current of a superluminescent diode. Our simultaneous suppression of thermal phase noise and relative intensity noise yields an angular random walk of 15 μdeg/√h and a bias instability of 33 μdeg/h using a fiber coil with a length of 5km and an effective area of 280m² for a measurement time of 40 hours.