An electronic device may have a housing such as a metal housing. A display may be mounted in the metal housing. Antenna structures may be mounted in the housing under an inactive peripheral portion of the display. Integrated circuits and other electrical components may be mounted in the housing under an active central portion of the display. Shielding structures may be configured to form a wall that extends between the display and the metal housing. The shielding structures may include a sheet of conductive fabric that is shorted to the metal housing and metal chassis structures in the display. The shielding structures may also include a tube of conductive fabric that is capacitively coupled to ground traces in a touch sensor panel. The conductive fabric tube and the sheet of conductive fabric may be shorted to each other using conductive adhesive.