Is it normal for students to ask for a lot of reference letters?
My solution as a student and my preferred solution as a prof who might get a lot of requests from a student would just be a heads up that a number of requests will be coming in the future months. 

That lets the writer know that it is good to keep copies on file, or even prepare a template file for this student so that it takes little time or effort to send out the next letter. Some professors will write a general letter to send everywhere and some will tailor it for a given institution. As the prof, I would probably do a bit of both, depending on whether I had a relationship with the receiving institution and on whether the student is specially qualified for a position. 

Don't make every request a surprise with a short deadline. That could be annoying.