The formation of small droplets and bubbles in turbulent flows is a crucial process in geophysics and engineering, whose underlying physical mechanism remains a puzzle. In this Letter, we address this problem by means of high-resolution numerical simulations, comparing a realistic multiphase configuration with a numerical experiment in which we attenuate the presence of strong velocity gradients either across the whole mixture or in the disperse phase only. Our results show unambiguously that the formation of small droplets is governed by the internal dynamics which occurs during the breakup of large drops and that the high vorticity and the extreme dissipation associated to these events are the consequence and not the cause of the breakup.