  Given a graph $H$, the Tur\'an number $ex(n,H)$ is the largest number of
edges in an $H$-free graph on $n$ vertices. We make progress on a recent
conjecture of Conlon, Janzer, and Lee on the Tur\'an numbers of bipartite
graphs, which in turn yields further progress on a conjecture of Erd\H{o}s and
Simonovits. Let $s,t,k\geq 2$ be integers. Let $K_{s,t}^k$ denote the graph
obtained from the complete bipartite graph $K_{s,t}$ by replacing each edge
$uv$ in it with a path of length $k$ between $u$ and $v$ such that the $st$
replacing paths are internally disjoint. It follows from a general theorem of
Bukh and Conlon that $ex(n,K_{s,t}^k)=\Omega(n^{1+\frac{1}{k}-\frac{1}{sk}})$.
Conlon, Janzer, and Lee recently conjectured that for any integers $s,t,k\geq
2$, $ex(n,K_{s,t}^k)=O(n^{1+\frac{1}{k}-\frac{1}{sk}})$. Among many other
things, they settled the $k=2$ case of their conjecture. As the main result of
this paper, we prove their conjecture for $k=3,4$. Our main results also yield
infinitely many new so-called Tur\'an exponents: rationals $r\in (1,2)$ for
which there exists a bipartite graph $H$ with $ex(n, H)=\Theta(n^r)$, adding to
the lists recently obtained by Jiang, Ma, Yepremyan, by Kang, Kim, Liu, and by
Conlon, Janzer, Lee. Our method builds on an extension of the Conlon-Janzer-Lee
method. We also note that the extended method also gives a weaker version of
the Conlon-Janzer-Lee conjecture for all $k\geq 2$.
