Kleene's Theorem and TMs
We can prove the following theorem:
Theorem: A language $L$ is regular if and only if there exists a DFA or NFA for language $L$.
Turing machines are more powerful than DFAs and NFAs. In particular,  every finite state automaton can be simulated by some Turing machine.
So, according to the above propositions, the answer to your question is yes: every regex has a Turing machine whose language is identical.