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Abstract Thwaites Glacier is one of the fastest‐changing ice‐ocean systems in Antarctica. Basal melting beneath Thwaites' floating ice shelf, especially around pinning points and at the grounding line, sets the rate of ice loss and Thwaites' contribution to global sea‐level rise. The rate of basal melting is controlled by the transport of heat into and through the ice–ocean boundary layer toward the ice base. Here we present the first turbulence observations from the grounding line of Thwaites Eastern Ice Shelf. We demonstrate that contrary to expectations, the turbulence‐driven vertical flux of heat into the ice–ocean boundary layer is insufficient to sustain the basal melt rate. Instead, most of the heat required must be delivered by lateral fluxes driven by the large‐scale advective circulation. Lateral processes likely dominate beneath the most unstable warm‐cavity ice shelves, and thus must be fully incorporated into parameterizations of ice shelf basal melting.