Increased ice velocities in Greenland are contributing significantly to eustatic sea level rise. Faster ice flow has been associated with ice-ocean interactions in water-terminating outlet glaciers and with increased surface meltwater supply to the ice-sheet bed inland. Observed correlations between surface melt and ice acceleration have raised the possibility of a positive feedback in which surface melting and accelerated dynamic thinning reinforce one another, suggesting that overall warming could lead to accelerated mass loss. Here I show that it is not simply mean surface melt but an increase in water input variability that drives faster ice flow. Glacier sliding responds to melt indirectly through changes in basal water pressure, with observations showing that water under glaciers drains through channels at low pressure or through interconnected cavities at high pressure. Using a model that captures the dynamic switching between channel and cavity drainage modes, I show that channelization and glacier deceleration rather than acceleration occur above a critical rate of water flow. Higher rates of steady water supply can therefore suppress rather than enhance dynamic thinning, indicating that the melt/dynamic thinning feedback is not universally operational. Short-term increases in water input are, however, accommodated by the drainage system through temporary spikes in water pressure. It is these spikes that lead to ice acceleration, which is therefore driven by strong diurnal melt cycles and an increase in rain and surface lake drainage events rather than an increase in mean melt supply.