Electronic devices may be provided with display structures such as glass and polymer layers in a liquid crystal display. The glass layers may serve as substrates for components such as a color filter layer and thin-film transistor layer. The polymer layers may include films such as a polarizer film and other optical films. During fabrication of a display, the polymer layers and glass layers may be laminated to one another. Portions of the polymer layers may extend past the edges of the glass layers. Laser cutting techniques may be used to trim away excess portions of the polymer layer that do not overlap underlying portions of the glass layers. Laser cutting may involve application of an adjustable infrared laser beam.