An electronic device housing may have conductive walls and planar metal internal housing structures. A display in the electronic device may have a touch sensor layer such as an array of capacitive touch sensor electrodes. The display may have display layers in a plastic chassis structure and a metal chassis structure. A display layer may have a conductive surface. Conductive tape may be used to couple the touch sensor to the metal chassis structure and a metal radio-frequency shielding structure. Conductive tape may also be used to short the conductive surface of the display layer to the internal metal housing structure. Segments of conductive tape may run along each of the four edges of a display and may overlap at display corners. Conductive foam structures may be used to short the segments to adjacent portions of the conductive housing walls.