A foldable display may have a display cover layer and a flexible display panel. The foldable display may bend around a bend axis. The display panel may have an array of pixels configured to display an image through the display cover layer. The display cover layer may be formed from a layer of glass. A recess may be formed in the layer of glass that extends along the bend axis. The recess may form a flexible locally thinned portion in the layer of glass that allows the glass layer to bend about the bend axis. To ensure that the display cover layer exhibits satisfactory impact resistance during drop events, corner portions of the display cover layer and other edge portions of the display cover layer may be provided with greater thickness relative to other portions of the display cover layer outside of the locally thinned portion.