What can I use instead of cornstarch in Turkish Delight?
Both the corn starch and cream of tartar are used for thickening in this case. You could try potato or tapioca starch; from my own experience and experiments, however, potato starch creates a very unusual, stretchy consistency. Tapioca starch would most likely produce better results than potato starch.

As for the cream of tartar, you can substitute each teaspoon of cream of tartar for three teaspoons of vinegar or lemon juice, according to this other Stack Exchange question.