Turing machine with "move to origin" instead of "move left"
When you want to move one position to the left, execute the following algorithm:


Mark the current position as special.
Move to the origin.
Translate the tape one cell to the right, while keeping the special marker in place.
Move to the origin.
Scan until reaching the special marker.
If you move to the right, don't forget to erase the special marker.


(This simulates a Turing machine with a single two-sided tape.)