A foldable electronic device may bend about a bend axis. A foldable display panel may be configured to bend along the bend axis as the foldable device is folded. A display cover layer may overlap the display panel. The display panel may have an array of pixels configured to display an image viewable 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 runs parallel to the bend axis and that overlaps the bend axis. The recess forms a flexible locally thinned portion in the glass that allows the display cover layer to bend. A rough surface texture and polymer coating may be provided in the recess. The recess may have shallowly sloping walls.