Graphene is considered a record-performance nonlinear-optical material on the basis of numerous experiments. The observed strong nonlinear response ascribed to the refractive part of graphene's electronic third-order susceptibility χ(3) cannot, however, be explained using the relatively modest χ(3) value theoretically predicted for the 2D material. Here we solve this long-standing paradox and demonstrate that, rather than χ(3)-based refraction, a complex phenomenon which we call saturable photoexcited-carrier refraction is at the heart of nonlinear-optical interactions in graphene such as self-phase modulation. Saturable photoexcited-carrier refraction is found to enable self-phase modulation of picosecond optical pulses with exponential-like bandwidth growth along graphene-covered waveguides. Our theory allows explanation of these extraordinary experimental results both qualitatively and quantitatively. It also supports the graphene nonlinearities measured in previous self-phase modulation and self-(de)focusing (Z-scan) experiments. This work signifies a paradigm shift in the understanding of 2D-material nonlinearities and finally enables their full exploitation in next-generation nonlinear-optical devices.