Until recently, the generation of super-octave continua in bulk materials at full repetition rates of femtosecond (fs) oscillators has been limited to a few special cases of Kerr lens mode-locked Ti:sapphire lasers. In 2019, we described a 3.4-octave fs source with 50 nJ pulse energy at the repetition rate of 78 MHz based on polycrystalline Cr:ZnS [Vasilyev et al., Optica6, 126 (2019)]. Here we explain the mechanism of fs supercontinuum generation in transition-metal doped polycrystalline II-VI semiconductors at relatively low (nJ-level) pulse energy. We demonstrate that this new supercontinuum regime is enabled by a complex, yet well-reproducible, interplay between the effects arising from the third-order nonlinearity, the quadratic nonlinearities, and thermal optical effects in the medium. We also demonstrate the control of fs pulse propagation in disordered χ(2) media via the control of the material microstructure.