We investigate power-scaling of a Kerr lens mode-locked (KLM) Yb:Lu_2O_3 thin-disk laser (TDL) oscillator operating in the sub-100-fs pulse duration regime. Employing a scheme with higher round-trip gain by increasing the number of passes through the thin-disk gain element, we increase the average power by a factor of two and the optical-to-optical efficiency by a factor of almost three compared to our previous sub-100-fs mode-locking results. The oscillator generates pulses with a duration of 95 fs at 21.1 W average power and 47.9 MHz repetition rate. We discuss the cavity design for continuous-wave and mode-locked operation and the estimation of the focal length of the Kerr lens. Unlike to usual KLM TDL oscillators, an operation at the edge of the stability zone in continuous-wave operation is not required. This work shows that KLM TDL oscillators based on the gain material Yb:Lu_2O_3 are an excellent choice for power-scaling of laser oscillators in the sub-100-fs regime, and we expect that such lasers will soon operate at power levels in excess of hundred watts.