In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links–links whose Khovanov homology is supported on two adjacent diagonals–are known to only contain torsion of order 2. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported in 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 torsion of order 2 over that range of homological gradings. These conditions are then shown to be met by an infinite family of 3-braids, strictly containing all 3-strand torus links, thus giving a partial answer to Sazdanovic and Przytycki’s conjecture that 3-braids have only torsion of order 2 in Khovanov homology. We also give explicit computations of integral Khovanov homology for all links in this family.