  The Turaev genus of a link can be thought of as a way of measuring how
non-alternating a link is. A link is Turaev genus zero if and only if it is
alternating, and in this viewpoint, links with large Turaev genus are very
non-alternating. In this paper, we study Turaev genus one links, a class of
links which includes almost alternating links. We prove that the Khovanov
homology of a Turaev genus one link is isomorphic to $\mathbb{Z}$ in at least
one of its extremal quantum gradings. As an application, we compute or nearly
compute the maximal Thurston Bennequin number of a Turaev genus one link.
