Ocean-driven submarine basal melt of the Greenland Ice Sheet remains a major source of uncertainty in climate projections of future sea level rise based on ice sheet modeling. State-of-the-art parameterizations focus on the properties of the ocean water reaching the marine-terminating glaciers, processes at the ice-ocean interface and subglacial discharge to estimate submarine basal melt. This study uses 2-dimensional, nonrotating, high-resolution ocean simulations complemented by theory and observations to quantify two effects of fjord bathymetry (sills) on the thermal forcing with consequences for the basal melt. These effects are the recirculation (reflux) of glacially modified water due to hydraulic control at the sill (which we study in detail) and the cooling of the inflow due to the restriction of the deepest and warmest Atlantic water inflow. For a fixed sill depth, an increase in subglacial discharge can make the exchange flow hydraulically controlled. Our results suggest that basal melt parameterizations should account for fjord bathymetry, and outline a road map to guide future parameterization developments relevant to narrow fjords.