Is it illegal to put a major brand on a shirt you customized?

It depends on what you mean. If you mean is it illegal to design a shirt then put a major brand label's logo on your shirt, for instance, the Coca-Cola brand logo, yes, that is illegal. Almost all, if not all, major brand labels are copyrighted and it is illegal to use them without permission.

If you mean can you make a shirt and call it a brand name that is the same a brand that already exists, for instance, American Eagle, that is illegal for the same reason.