  Do\v{s}li\'{c} et al.~defined the Mostar index of a graph $G$ as
$\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$,
the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller
distance in $G$ to $u$ than to $v$. Contributing to conjectures posed by
Do\v{s}li\'{c} et al., we show that the Mostar index of bipartite graphs of
order $n$ is at most $\frac{\sqrt{3}}{18}n^3$, and that the Mostar index of
split graphs of order $n$ is at most $\frac{4}{27}n^3$.
