Variant of Turing machine
In order to prove that your modified Turing machine is equivalent to a standard Turing machine, you need to do two things:


Given a standard Turing machine $T_1$, give a modified Turing machine $T_2$ such that $L(T_1) = L(T_2)$.
Given a modified Turing machine $T_3$, give a standard Turing machine $T_4$ such that $L(T_3) = L(T_4)$.


Good luck!