Europa, the innermost icy satellite of Jupiter, has a tortured young surface and sustains a liquid water ocean below an ice shell of highly debated thickness. Quasi-circular areas of ice disruption called chaos terrains are unique to Europa, and both their formation and the ice-shell thickness depend on Europa's thermal state. No model so far has been able to explain why features such as Conamara Chaos stand above surrounding terrain and contain matrix domes. Melt-through of a thin (few-kilometre) shell is thermodynamically improbable and cannot raise the ice. The buoyancy of material rising as either plumes of warm, pure ice called diapirs or convective cells in a thick (>10 kilometres) shell is insufficient to produce the observed chaos heights, and no single plume can create matrix domes. Here we report an analysis of archival data from Europa, guided by processes observed within Earth's subglacial volcanoes and ice shelves. The data suggest that chaos terrains form above liquid water lenses perched within the ice shell as shallow as 3kilometres. Our results suggest that ice-water interactions and freeze-out give rise to the diverse morphologies and topography of chaos terrains. The sunken topography of Thera Macula indicates that Europa is actively resurfacing over a lens comparable in volume to the Great Lakes in North America.