Microlensing by the stellar population of lensing galaxies provides an important opportunity to resolve the accretion disc structure spatially in strongly lensed quasars. Disc sizes estimated in this way are on average larger than the predictions of the standard Shakura–Sunyaev accretion disc model. An analysis of the observational data on microlensing variability suggests that some fraction of lensed quasars (primarily smaller-mass objects) are accreting in the super-Eddington regime. Super-Eddington accretion leads to the formation of an optically thick envelope scattering the radiation formed in the disc. This makes the apparent disc size larger and practically independent of wavelength. In the framework of our model, it is possible to make self-consistent estimates of mass accretion rates and black hole masses for the cases when both amplification-corrected fluxes and radii are available.