OPEN

Show that for any rational $\alpha \in (1,2)$ there exists a bipartite graph $G$ such that
\[\mathrm{ex}(n;G)\asymp n^{\alpha}.\]
Conversely, if $G$ is bipartite then must there exist some rational $\alpha$ such that\[\mathrm{ex}(n;G)\asymp n^{\alpha}?\]

A problem of Erdős and Simonovits.