An electronic device may have a display overlapped by a display cover layer. Portions of the surface of the display and cover layer may have curved profiles. For example, a display cover layer may have transparent sidewall portions with curved surface profiles. The transparent sidewall portions of the display cover layer may include rounded comers having areas of compound curvature. A flexible display panel may be pressed over a mold to impart desired curvature (such as compound curvature) to the flexible display panel. To mitigate wrinkling in a flexible display panel molded to have compound curvature, a backfilm may be included that absorbs compressive strains in the display panel. The backfilm may have a coefficient of thermal expansion that is higher than that of the display panel. Instead or in addition, the backfilm may have portions with different Young's modulus magnitudes or different thickness magnitudes.