An electronic device may have a display with a bent portion such as one or more bent edge portions. The display may be formed from a flexible display layer such as an organic light-emitting-diode layer. The organic light-emitting-diode layer may include a substrate layer such as a sheet of polymer. The flexible display may be attached to a support structure such as a flexible support layer using adhesive. To facilitate bending, the display may be heated. Bending and heating equipment may simultaneously heat the display layer and the flexible support layer to form a bent edge portion for the display. The bending and heating equipment may include heated structures for pressing the display layer into a desired shape.