A window such as a vehicle window may have an infrared-light-blocking area formed from an infrared-light-blocking layer. The window may also have an infrared-light-transparent area formed from an infrared-light-transparent layer that is not covered by the infrared-light-blocking layer. The infrared-light-transparent layer may have a conductive layer that is transparent to near-infrared light such as a conductive oxide layer or a metal layer. The infrared-light-blocking layer may have one or more layers of infrared-light-reflecting material such a silver layer that is electrically coupled to the conductive layer in the infrared-light-transmitting layer. Ohmic heating current may be passed to the conductive layer in the infrared-light-transmitting layer through the conductive layer in the infrared-light-blocking layer or through elongated metal contacts that run along edges of the infrared-light-transmitting layer.