Although the bulk of the observed optical flux from the discs of intermediate-redshift lensed quasars is formed well outside the region of strong relativistic boosting and light bending, relativistic effects have an important influence on microlensing curves. The reason lies in the divergent nature of amplification factors near fold caustics, which are increasingly sensitive to small spatial size details. Higher-order disc images produced by strong light bending around the black hole may affect the amplification curves, making a contribution of up to several per cent near maximum amplification. In accordance with theoretical predictions, some of the observed high-amplification events possess fine structure. Here we consider three putative caustic-crossing events, one by SBS J1520+530 and two events for individual images of Einstein’s cross (QSO J2237+0305). Using relativistic disc models allows us to improve the fits but the required inclinations are high, . Such high inclinations apparently contradict the absence of any strong absorption that is likely to arise if a disc is observed edge-on through a dust torus. Still, high inclinations are required only for the central parts of the disc, which allows the disc itself initially to be tilted by 60–90° with respect to the black hole and aligned toward the black hole equatorial plane near the last stable orbit radius. For SBS J1520+530, an alternative explanation for the observed amplification curve is a superposition of two subsequent fold-caustic crossings. While relativistic disc models favour black hole masses ∼1010 M⊙ (several times higher than the virial estimates) or small Eddington ratios, this model is consistent with the observed distribution of galaxies over peculiar velocities only if the black hole mass is .