The geological nature of glacier beds plays a key role in ice sheet dynamics. Whereas little is known about Greenland’s subglacial geology, the presence of basal sediments is a necessary condition for fast-flowing Antarctic ice streams. Such sediments sustain subglacial till layers, which if water saturated and under high pore-water pressure provide little resistance to ice flow. Using receiver function modeling of teleseismic P-waves, we report a thick (at least tens of meters) sediment layer beneath a site in Greenland’s ablation zone, ∼15 km away from the western ice sheet margin. Although we do not discuss the origin or detailed nature of these subglacial sediments, we suggest that they are capable of sustaining a till layer. Due to the prevalence of an inefficient, pressurized subglacial drainage system, this till layer would typically be under high pore pressures and fail at the shear stresses transmitted from the overlaying ice. Ice flow resistance is thus focused on regions where till is consolidated or absent, or where form drag over obstacles takes place. In contrast to Antarctica, Greenland’s surface melt now affects practically all latitudes, and efficient hydraulic connections between ice sheet surface and bed are pervasive throughout the ablation zone. This implies that rapid, melt-induced changes in subglacial water pressure are possible. The time required for basal till strength to react to these changes depends on the till’s properties. We estimate that formation and destruction of flow resistance can occur on time scales of less than a few years. This could lead to changes in ice flow that are currently difficult to predict.