Do American universities have higher psychology degrees besides bachelor's degrees?

Psychology is psychology. In addition to bachelor's degrees, there are also masters and doctors. The master of psychology is usually MS (Master of Science), or there is also a doctor of MA (Master of Arts), usually PhD (Doctor of Philosophy). But some schools also have a kind of PsyD (doctor of psychology). If it is related to education, you can also take the EdD (doctor of education).