An electronic device may be provided with an electronic component such as a camera light source containing a light-emitting diode. During operation, the light-emitting diode may produce heat. Thermally conducting elastomeric structures may have features such as sidewalls that mate with external surface of the camera light source or other electronic component to dissipate heat from the electronic component. Metal structures such as a bracket may be used to press the elastomeric structures and the electronic component towards a wall of a housing for the electronic device. Support structures may be interposed between the wall of the housing and the elastomeric structures. The support structures may have an opening that is aligned with an opening in the housing wall. Insert structures may be received within the opening in the support structures. The electronic component may be aligned with the insert structures and the opening in the support structures.