Electronic devices may be provided with electronic components and cowling structures that secure the electronic components. A cowling structure may include a metal portion and an insulating portion that has been insert-molded onto the metal portion. The metal portion and the insert-molded insulating portion may each have an opening that receives a screw. The screws may pass through the respective opening and attach to a substrate. The substrate may be a transparent cover layer for a device display. The cowling structure may press the electronic components against the transparent substrate layer. The device may include an antenna. The insert-molded insulating portion may extend from an edge of the metal portion in the direction of the antenna. The insert-molded insulating portion may prevent one of the screws from forming an electrical connection with the metal portion.