Can we be registered in another university when we are already registered in a university in Turkey?

Yes, in Turkey after your studies a year in university you can switch to another public or private university the next year.  However, it is not a very easy process and the conditions to move from private university to state university are really hard.  Your GPA must be very high.  Transition from private university to another private university is easier.  In the first year, you can study language education at one private university and your department in another  private university.  In this case, you must pass the Language exam of the second university.