  Given a graph $H$, the extremal number $\mathrm{ex}(n,H)$ is the largest
number of edges in an $H$-free graph on $n$ vertices. We make progress on a
number of conjectures about the extremal number of bipartite graphs. First,
writing $K'_{s,t}$ for the subdivision of the bipartite graph $K_{s,t}$, we
show that $\mathrm{ex}(n, K'_{s,t}) = O(n^{3/2 - \frac{1}{2s}})$. This proves a
conjecture of Kang, Kim and Liu and is tight up to the implied constant for $t$
sufficiently large in terms of $s$. Second, for any integers $s, k \geq 1$, we
show that $\mathrm{ex}(n, L) = \Theta(n^{1 + \frac{s}{sk+1}})$ for a particular
graph $L$ depending on $s$ and $k$, answering another question of Kang, Kim and
Liu. This result touches upon an old conjecture of Erd\H{o}s and Simonovits,
which asserts that every rational number $r \in (1,2)$ is realisable in the
sense that $\mathrm{ex}(n,H) = \Theta(n^r)$ for some appropriate graph $H$,
giving infinitely many new realisable exponents and implying that $1 + 1/k$ is
a limit point of realisable exponents for all $k \geq 1$. Writing $H^k$ for the
$k$-subdivision of a graph $H$, this result also implies that for any bipartite
graph $H$ and any $k$, there exists $\delta > 0$ such that
$\mathrm{ex}(n,H^{k-1}) = O(n^{1 + 1/k - \delta})$, partially resolving a
question of Conlon and Lee. Third, extending a recent result of Conlon and Lee,
we show that any bipartite graph $H$ with maximum degree $r$ on one side which
does not contain $C_4$ as a subgraph satisfies $\mathrm{ex}(n, H) = o(n^{2 -
1/r})$.
