During grid faults, grid-forming converters (GFC) switch to current limiting operation, which can lead to uncontrollable power output and potential transient instability. To ensure GFC can meet the requirements of current limiting and maintaining synchronization stability with the system during faults, while also addressing transient overcurrent issue following fault clearance, an adaptive virtual impedance control strategy is proposed to improve transient stability. First, based on the analysis of power-angle curve during GFC transient operation, the transient stability conditions for GFC are established. Then, the impact of virtual impedance on fault current limiting and synchronization stability during transients is analyzed. An adaptive virtual inductance closed-loop control strategy is proposed to enhance synchronization stability, while an adaptive virtual resistance is employed to limit fault overcurrent. Finally, hardware- in-the-loop tests on a single-machine infinite-bus system and simulations on an 8-bus system are conducted to validate the effectiveness of the proposed control strategy. Results show that the strategy effectively limits fault overcurrent, maintains stable power output during faults, and avoids peak power and current spikes upon fault recovery.