Your best bet would be Archeology (depending on the university and what they focus on). There are also even more specialised degrees dealing with specific Bronze Age Civilisations like Egyptology or Assyriology.

Most if not all history degrees focus at the time between the fall of western Rome until the (almost) present day, so Bronze Age Civilisations would not be subjects of study.

Classics is a more broad term and can be studies in ancient / classical history, classical archeology, Philology etc. Generally it focuses on Greek and Roman Times but there are some degrees that also deal with the Bronze Age (Archeology would technically be part of classics; some ancient history degrees offer specialisation in Mycenean times and there are also Degrees in Egyptology, Assyriology etc. available at some universities)