An electronic device may have structures that are coupled together using conductive adhesive such as anisotropic conductive film and other adhesives. The structures that are coupled together may include a touch sensor structure formed from electrodes on the inner surface of a display cover layer, a display module having display layers such as a thin-film transistor layer, and circuitry mounted on substrates such as printed circuits. Conductive signal path structures may be used in routing signals within the electronic device. The conductive signal path structures may be formed from pins that are embedded within injection molded plastic, from metal traces such as laser-deposited metal traces that are formed on the surface of a plastic member or other dielectric, from metal structures that run within channels in a plastic, printed circuit traces, and other signal path structures.