What does sublinear space mean for Turing machines?
When dealing with restricted space, we use the following model. The Turing machine has three tapes: a read-only input tape, a read-write work tape, and a write-only output tape. We only measure space consumption on the work tape. For palindromes, with space $O(\log n)$ on the work tape we can implement FOR loops that go over the input, comparing matching characters on both ends. Each index takes $O(\log n)$ space to store.