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SOLVED - $500
If $H$ is bipartite and is not $r$-degenerate, that is, there exists an induced subgraph of $H$ with minimum degree $>r$ then \[\mathrm{ex}(n;H) > n^{2-\frac{1}{r}+\epsilon}.\]
Conjectured by Erdős and Simonovits. Disproved by Janzer [Ja21], who constructed, for any $\epsilon>0$, a $3$-regular bipartite graph $H$ such that \[\mathrm{ex}(n;H)\ll n^{\frac{4}{3}+\epsilon}.\]

See also [113] and [146].

See also the entry in the graphs problem collection.

Additional thanks to: Zachary Hunter