Bottom-up synthesized GNRs and GNR heterostructures have promising electronic properties for high performance field effect transistors (FETs) and ultra-low power devices such as tunnelling FETs. However, the short length and wide band gap of these GNRs have prevented the fabrication of devices with the desired performance and switching behaviour. Here, by fabricating short channel (Lch ~20 nm) devices with a thin, high-k gate dielectric and a 9-atom wide (0.95 nm) armchair GNR as the channel material, we demonstrate FETs with high on-current (Ion >1 uA at Vd = -1 V) and high Ion/Ioff ~10^5 at room temperature. We find that the performance of these devices is limited by tunnelling through the Schottky barrier (SB) at the contacts and we observe an increase in the transparency of the barrier by increasing the gate field near the contacts. Our results thus demonstrate successful fabrication of high performance short-channel FETs with bottom-up synthesized armchair GNRs.