Counting rates of superconducting nanowire single-photon detectors are usually estimated at hundreds of MHz by their kinetic-inductive reset time. This maximum is also limited by capacitor coupling effects in conventional readout circuits. In this Letter, we design and demonstrate an improved readout circuit that reduces the reset time and removes circuit limits. The counting rate at the 3 dB compression point is increased by four times for a large active area detector. We also discuss nonlinear dependences of the counting rate on the incident continuous-wave optical power and give a numerical model to explain our observations.