The broadening of droplet size distributions in stratiform clouds, where the updraft velocity is almost zero, is puzzling. Without turbulence, the classical treatment of condensational growth of cloud droplets fails to explain this broadening. We investigated the time evolution of droplet size distributions using direct numerical simulations, where turbulence is resolved into the smallest scales. We found that the broadening is due to the turbulence-facilitated supersaturation fluctuations.