Wireless circuitry in an electronic device may contain radio-frequency transceiver circuitry and power amplifier circuitry that transmits radio-frequency signals through an antenna. A tap and power detector that are interposed in the radio-frequency signal path between the power amplifier circuitry and the antenna may be used to make output power measurements. Control circuitry may control the wireless circuitry in an open-loop control regime in which output power adjustments are based on a requested power without using the output power measurements. The control circuitry may also control the wireless circuitry in a closed-loop control regime in which the output power measurements serve as a source of real time feedback to determine whether to increase or decrease the output power. The output power may be controlled using linear transition zone power control curves in a transition zone between the open-loop regime and the closed-loop regime.