Turing machine has unbounded memory. There is a difference between unbounded and unlimited memory: You can extend the tape as needed, but at any given step of computation, the used space is finite.

A Turing machine with bounded memory is called linear bounded automaton. Although LBA is less powerful than a Turing machine in the absolute sense, it is not trivial to form a problem that a Turing machine could decide but a LBA could not.