A display such as a liquid crystal display may have an array of pixels that is illuminated using backlight illumination from a backlight. The backlight may have a light guide layer that distributes light from light-emitting diodes across the display. The light guide layer may have a planar portion that provides backlight illumination to the array of pixels and may have bent edge portions that curve out of the plane of the planar portion. Light scattering structures may be formed in the planar portion to extract backlight illumination from the light guide layer. A light sensor adjacent to the bent portion may monitor leaked light. The light guide layer may have two bent portions on opposing edges of the light guide layer.