Erdős Problem #113 — Bipartite graphs with an $n^{3/2}$ Turán bound

For a bipartite graph $H$, is $\operatorname{ex}(n,H)=O(n^{3/2})$ if and only if $H$ is $2$-degenerate, meaning every nonempty subgraph of $H$ has a vertex of degree at most $2$?