Fjords commonly punctuate continental edges formerly occupied by Quaternary ice sheets, reaching kilometre depths and extending many tens of kilometres inland. These features must have been created by late Cenozoic ice sheets, because rivers cannot erode bedrock much below sea level. Ice sheets drain primarily through fjords; therefore, widespread fjord insertion may have altered ice-sheet size, shape and dynamics. Here, we use a two-dimensional ice-sheet model to simulate the incision of fjords through a coastal mountain range. We show that topographic steering of ice and erosion proportional to ice discharge are sufficient to form fjords. Within one million years, kilometre-deep fjords punched through the mountain range owing to a robust positive feedback initiated by ice being steered towards mountain passes. Enhanced erosion beneath thicker, faster ice deepens these passes, amplifying the topographic steering. Simulated fjords are deepest through the highest topography and drain a large fraction of the interior ice. Ice sheets simulated on landscapes with existing fjords are generally smaller and exhibit longer response times and larger responses to climate changes, suggesting that modern ice sheets are more sensitive to climate fluctuations than Early Quaternary ice sheets.