Turing machine that accepts language with more a's than b's
Sit before a big empty sheet of paper. Write down a sequence of a's and b's. Now mechanically check whether that string should be accepted. You cannot use counting, that would be requiring an infinite number of "states" inside your head. 
You can however use additional symols. A striked out b is just a new symbol that is written on the tape. 

Turn your intuition into an algorithm. First informally, that is what Karolis has done. Then add information on how to implement this on a Turing tape. What direction is the head moving? Where will it stop? Then translate the whiles and untils into states and transitions (sort of goto-programming, which is improper unless you are programming a TM).

If you are a CS student, then you can do these steps. You can program. No matter whether it is C++, assembler, or Turing. That is just details.