Is it permissible to perform Istikhara prayer to find out whether a certain girl—whom a young man wishes to marry but cannot propose to for three years—is destined for him or not, in order to avoid becoming attached to her or living in false hope?

Istikhara is not for unveiling the unseen. Rather, it was legislated for the servant to entrust their Lord with choosing what is best for them when contemplating a permissible matter. If you are currently unable to marry, then turn away from the girl and occupy yourself with what benefits you. Then, perform Istikhara when you are able to marry. If you wish to inform the girl of your desire to marry and she agrees to wait, there is no objection, and Istikhara is legislated, provided there is no relationship between you two until the marriage contract.