  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 only contain $\mathbb{Z}_2$
torsion. 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 $\mathbb{Z}_2$
torsion 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 $\mathbb{Z}_2$ torsion in
Khovanov homology. We also give explicit computations of integral Khovanov
homology for all links in this family.
