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Supraglacial lakes play an important role in establishing hydrological connections that allow lubricating seasonal meltwater to reach the base of the Greenland Ice Sheet. Here we use new surface velocity observations to examine the influence of supraglacial lake drainages and surface melt rate on ice flow. We find large, spatially extensive speedups concurrent with times of lake drainage, showing that lakes play a key role in modulating regional ice flow. While surface meltwater is supplied to the bed via a geographically sparse network of moulins, the observed ice-flow enhancement suggests that this meltwater spreads widely over the ice-sheet bed. We also find that the complex spatial pattern of speedup is strongly determined by the combined influence of bed and surface topography on subglacial water flow. Thus, modeling of ice-sheet basal hydrology likely will require knowledge of bed topography resolved at scales (sub-kilometer) far finer than existing data (several km).