An electronic device may have a display mounted in a housing. The display may have a display panel with an array of pixels on a flexible substrate. A display cover layer may overlap the display panel. The flexible substrate may have a protruding portion that forms a tail. The tail may be coupled to a printed circuit on which a display driver integrated circuit and/or other circuitry is mounted. When the display is mounted in the housing, the tail may be bent back on itself to create a bend. The bend may be embedded in a molded polymer member. The molded polymer member may be attached to the housing with adhesive and may directly contact an inner surface of the display cover layer.