Turing machine where the next step is determined by the state and the symbols up to the read/write head

Question

Given a modified type of turning machine where

$\delta = Q\times \Gamma^* \implies Q\times \Gamma \times \{L,R\}$

where the next step of the machine is determined by the current state and whatever written on the tape up to the current point.

For example: if the tape content is $a\_ab\_\$\$a$, and the head is on the left \$, then the next step is determined by the state and the word $a\_ab\_\$$.

1. How can I prove that unrecognized languages can be recognized with these types of machines?
2. Do those machines contradict the Church–Turing thesis?