In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is, links whose Khovanov homology is supported on two adjacent diagonals, are known to contain only $\mathbb {Z}_2$ torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported on two adjacent diagonals over a range of homological gradings and the Khovanov homology satisfies some other mild restrictions, then the Khovanov homology of that link has only $\mathbb {Z}_2$ torsion over that range of homological gradings. These conditions are then shown to be met by an infinite family of three-braids, strictly containing all three-strand torus links, thus giving a partial answer to Sazdanović and Przytycki’s conjecture that three-braids have only $\mathbb {Z}_2$ torsion in Khovanov homology. We use these computations and our main theorem to obtain the integral Khovanov homology for all links in this family.