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.

Substituting for corn starch, you might look for corn flour (which might the same thing as corn starch depending on location), or look at a chart for substituting other starch thickeners, including rice starch, potato starch, tapioca starch, and others, which will have the closest equivalents. You might consider working with potato starch or rice starch, which are substituted on a one-to-one basis. For tapioca, you would need the same amount of tapioca starch, or about twice as much if using instant or quick tapioca (which will give a very different texture).

if you really can't find any of those, you might look into some thickeners that are further removed, like ground flaxseed, regular flour, tapioca, or arrowroot. If you can get your hands on arrowroot, it will give a similar glossy shine - you need an equal or slightly greater amount, and cook just a little longer than corn starch. would need about twice as much flour (2tb per tb), and make sure it's cooked very well - it is very stable, but won't be as glossy, and will have a raw flavor if under-cooked, so it isn't the best substitute - but it might do if you have absolutely nothing else. Rice flour might be very interesting, you might end up with something closer to mochi. You should be aware that using any of these substitutions will change the final product (the starches less so than other thickeners), and may be quite different - but the result might still be tasty, if you want to try it.

For cream of tartar (if you can find tartaric acid, it's the same thing), you might use an equal amount of white vinegar or lemon juice. The substitution will vary depending on what you're using it for, but I think it's probably more like stabilizing egg whites or use in frosting, rather than baking or use with baking soda, so equal amounts. You might even be able to omit it, since it is being used either to prevent crystallization of the sugars, or to stabilize the gelling process of the starches - you run the risk of a slightly less firm or creamy texture, depending.

Turkish delight is not just any sweetened gel, it is very specifically a gel made with starch, and using anything else will make a result which is not recognizable as Turkish delight.

If you cannot find corn starch,other starches will work quite well. Wheat or rice starch will probably be as good as cornstarch. I am not entirely sure about the waxy starches such as arrowroot. Any other thickeners will make a different dish.

The cream of tartar is only there for the acid, so use whatever culinary acid you have handy.

Rice starch is the usual substitute and of course is gluten free which seems important these days. Rice starch IS used in other parts of Asia I'm told to make a similar product. I've given it a try and I'd be pushed to tell the difference. But like anything else it's all about trial, error and experience I suppose.