Ice shelves in the Amundsen Sea Embayment have thinned, accelerating the seaward flow of ice sheets upstream over recent decades. This imbalance is caused by an increase in the ocean-driven melting of the ice shelves. Observations and models show that the ocean heat content reaching the ice shelves is sensitive to the depth of thermocline, which separates the cool, fresh surface waters from warm, salty waters. Yet the processes controlling the variability of thermocline depth remain poorly constrained. Here we quantify the oceanic conditions and ocean-driven melting of Cosgrove, Pine Island Glacier (PIG), Thwaites, Crosson, and Dotson ice shelves in the Amundsen Sea Embayment from 1991 to 2014 using a general circulation model. Ice-shelf melting is coupled to variability in the wind field and the sea-ice motions over the continental shelf break and associated onshore advection of warm waters in deep troughs. The layer of warm, salty waters at the calving front of PIG and Thwaites is thicker in austral spring (June–October) than in austral summer (December–March), whereas the seasonal cycle at the calving front of Dotson is reversed. Furthermore, the ocean-driven melting in PIG is enhanced by an asymmetric response to changes in ocean heat transport anomalies at the continental shelf break: melting responds more rapidly to increases in ocean heat transport than to decreases. This asymmetry is caused by the inland deepening of bathymetry and the glacial meltwater circulation around the ice shelf.