Turpentine makes a good airbrush cleaner. I have never used it to thin hobby paints.
Generic paint thinner, well, it depends on the paint you're using.
Lacquer thinner will thin almost any solvent based paint, from the Testors enamels, to Floquil, to Scalecoat, and so on, but will attack plastic, so you really have to know your stuff when using it on plastics.
Using the manufacturer's system is great for beginners, but as you learn, you will find out that the hobby paint business is very profitable because they take common items and put small quantities in small bottles and sell them for a large profit . Nothing wrong with a little capitalism, but then there's nothing wrong with saving a little money either
Xylene or Xylol will thin most hobby enamels, and is not as volatile as lacquer thinner. I have used it for years without any problems at all. I have done many custom paint jobs for paying customers and nobody as ever complained
You should test the thinner you want to try. Fill a small container with it, and dip a toothpick in your paint, and in the proposed thinner. If the paint dissolves into the thinner, you're good. If it stays on the toothpick, or breaks off and floats in the thinner like a little booger, then don't use it.
For acrylics, some thin with alcohol, some with water, and some with windex. Again, test it. Tamiya paints use an alcohol based thinner. Others use either their own, or Windex, or even water. I have not had much luck with Polly S. The paint is very thick. Pollyscale is better. Badger Modelflex is also good and formulated for spraying. No thinning is necessary with it. One place I do stick with the system is retarders. These are necessary with some acrylics because they tend to dry very fast. With Modelflex I add a squirt of their retarder to every new bottle. They airbrush better that way.
These methods work for me. There are others, and you will get other opinions. Good Luck!
PS, make sure you use a proper respirator with the solvents mentioned, even if you have a vented booth, and I do mean a respirator, not a dust mask!
Modeling Espee on the Coast and in steam