Answer:
The princess can perform a juggling. In this process, the three apples take turns as they are thrown up in the air and received back from where they are being thrown. In a perfect juggling situation where three balls (apples in this case are involved), there is no point in time where the three balls are all in the hands of the player. Doing this, the princess should make sure that at every point in time when crossing the bridge, not more than two of the apples are in her hands and the third apple in the air. Though a risky game it poses itself as one of the possible ways out of this dilemma.
Hope this helps!