We have experimentally investigated supercontinuum (SC) generation and the evolution of optical damage in sapphire and YAG crystals with 180 fs, 1035 nm pulses from an amplified Yb:KGW laser with variable repetition rates, both in tight and loose focusing conditions. In this Letter, we demonstrate that the extinction of the SC spectrum always correlates with an occurrence of conical third harmonic generation, which readily serves as an indication of the onset of in-bulk optical damage. Damage-related structural changes of the nonlinear material are also justified by an increased intensity and large red shift of crystal luminescence spectrum corresponding to the F center emission. The SC spectrum in sapphire starts shrinking on the time scale between seconds and minutes by varying the focusing condition from tight to loose at the laser repetition rate of 200 kHz, whereas the YAG crystal produces stable performance for several hours at least.