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Rapid, large amplitude variability at optical to X-ray wavelengths is now seen in an increasing number of Seyfert galaxies and luminous quasars. The variations imply a global change in accretion power, but are too rapid to be communicated by inflow through a standard thin accretion disc. Such discs are long known to have difficulty explaining the observed optical/UV emission from active galactic nuclei. Here we show that alternative models developed to explain these observations have larger scale heights and shorter inflow times. Accretion discs supported by magnetic pressure in particular are geometrically thick at all luminosities, with inflow times as short as the observed few year time-scales in extreme variability events to date. Future time-resolved, multiwavelength observations can distinguish between inflow through a geometrically thick disc as proposed here, and alternative scenarios of extreme reprocessing of a central source or instability-driven limit cycles.