In order to characterize and calibrate quantum processing devices, a large amount of measurement data has to be collected. Active qubit reset increases the speed at which data can be gathered but requires additional hardware and/or calibration. The experimental apparatus can, however, be operated at elevated repetition rates without reset. In this case, the outcome of a first measurement serves as the initial state for the next experiment. Rol et al. have used this restless operation mode to accelerate the calibration of a single-qubit gate by measuring fixed-length sequences of Clifford gates that compose to X gates [Phys. Rev. Appl. 7, 041001 (2017)]. However, we find that when measuring pulse sequences that compose to arbitrary operations, distortion appears in the measured data. Here, we extend the restless methodology by showing how to efficiently analyze restless measurements and correct distortions to achieve an identical outcome and accuracy as compared to measurements in which the superconducting qubits are reset. This allows us to rapidly characterize and calibrate qubits. We illustrate this bias-corrected restless measurements method by measuring a Rabi oscillation at a 250 kHz repetition rate without any reset, for a qubit with a decay rate of $ 1/2π T_1 $ = 3 kHz. We also show that we can measure a single- and a two-qubit average ate fidelity using randomized benchmarking 20 and 8 times faster, respectively, than measurements that reset the qubits through $ T_1 $ decay.