Using time-resolved luminescence we measure the energy relaxation of excitons in thin quantum wells. We find that the exciton relaxation in thin wells cannot be explained by the models of exciton relaxation generally used in thick wells. Instead, the excitons relax by losing potential energy in a drift-diffusion motion driven by potential fluctuations in the quantum-well plane, which are longer than the coherence length of the excitons. The well-width dependence of the exciton relaxation and differences between quantum wells with and without growth interruption are discussed. © 1991 The American Physical Society.