Turan numbers of bipartite subdivisions

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 $k\geq 2$.