How is the number of states in a Turing machine bounded? The definition of Turing machine says that the number of states is finite. However, I do not get how this can be true. Is the number of states in a Turing machine actually not fixed, that is not bounded? If not, can anyone explain how this is possible?
You seem to have misunderstood the definition of a Turing machine. The "state" refers not to the whole configuration of the machine (tape contents, position of the head, etc.) but to the state of the finite automaton that controls the tape head.  The automaton has a finite number of states by definition.