Magnetic fields perpendicular to superconducting films often trigger vortex avalanches, which always are very harmful for electronic devices and other applications. Such avalanches can be suppressed by a metal layer placed in contact with the superconductor surface, an effect that up to now has been thought to be a consequence of improved heat conduction. Here we show experimentally that the role of the metal layer is not that of a heat-sink, but rather that of an electromagnetic drag due to eddy currents induced in the metal layer during the abrupt onset of the flux avalanches. The effect is demonstrated for films of MgB2 and Nb.