We explore the spatio-temporal evolution of solar flares by fitting a radial expansion model r(t) that consists of an exponentially growing acceleration phase, followed by a deceleration phase that is parameterized by the generalized diffusion function r(t){prop.to}{kappa}(t-t_1_)^{beta}/2^, which includes the logistic growth limit ({beta}=0), sub-diffusion ({beta}=0-1), classical diffusion ({beta}=1), super-diffusion ({beta}=1-2), and the linear expansion limit ({beta}=2). We analyze all M- and X-class flares observed with Geostationary Operational Environmental Satellite and Atmospheric Imaging Assembly/Solar Dynamics Observatory (SDO) during the first two years of the SDO mission, amounting to 155 events. We find that most flares operate in the sub-diffusive regime ({beta}=0.53+/-0.27), which we interpret in terms of anisotropic chain reactions of intermittent magnetic reconnection episodes in a low plasma-{beta} corona. We find a mean propagation speed of v=15+/-12km/s, with maximum speeds of v_max_=80+/-85km/s per flare, which is substantially slower than the sonic speeds expected for thermal diffusion of flare plasmas. The diffusive characteristics established here (for the first time for solar flares) is consistent with the fractal-diffusive self-organized criticality model, which predicted diffusive transport merely based on cellular automaton simulations.