Electronic devices may be provided having internal components mounted to a structural glass support member. The structural glass support member may have a planar front surface that forms a front surface of the device. The structural glass support member may have bent portions that form sidewall surfaces of the device. Portions of the structural glass support member may form a transparent display cover layer. A rigid or flexible display may be mounted to the structural glass support member. Additional internal device components may be mounted to the display. A thin enclosure for enclosing the internal components in the device may be mounted to the structural glass support member. The thin enclosure may be mounted to the structural glass support member using a peripheral member. The thin enclosure may be free from attachments to internal components or may be adhesively bonded to one or more internal components.