An electronic device may have a housing that separates an exterior region from an interior region. The housing may have a front layer on a front face of the housing and a rear layer on an opposing rear face of the housing. Sidewall structures may extend between the front and rear layers. The housing may form a head-mounted housing that is configured to be worn on a user's head. An internal frame may be mounted in the interior region. The internal frame may have a nose bridge structure that is coupled to the housing with a coupling member such as a coupling member formed from an elastomeric vibration damping material. Other portions of the frame such elongated laterally extending support members may not contact any portion of the housing and may therefore be isolated from the housing during drop events.