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Given a graph $H$, the Tur\'an number $ex(n,H)$ is the largest number ofedges in an $H$-free graph on $n$ vertices. We make progress on a recentconjecture of Conlon, Janzer, and Lee on the Tur\'an numbers of bipartitegraphs, which in turn yields further progress on a conjecture of Erd\H{o}s andSimonovits. Let $s,t,k\geq 2$ be integers. Let $K_{s,t}^k$ denote the graphobtained 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 ofBukh 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 $s,t,k\geq2$, $ex(n,K_{s,t}^k)=O(n^{1+\frac{1}{k}-\frac{1}{sk}})$. Among many otherthings, they settled the $k=2$ case of their conjecture. As the main result ofthis paper, we prove their conjecture for $k=3,4$. Our main results also yieldinfinitely many new so-called Tur\'an exponents: rationals $r\in (1,2)$ forwhich there exists a bipartite graph $H$ with $ex(n, H)=\Theta(n^r)$, adding tothe lists recently obtained by Jiang, Ma, Yepremyan, by Kang, Kim, Liu, and byConlon, Janzer, Lee. Our method builds on an extension of the Conlon-Janzer-Leemethod. We also note that the extended method also gives a weaker version ofthe Conlon-Janzer-Lee conjecture for all $k\geq 2$.

Turan numbers of bipartite subdivisions