Given a graph $H$ and a positive integer $n$ , the Turán number $\mathrm{ex}(n,H)$ is the maximum number of edges in an $n$ -vertex graph that does not contain $H$ as a subgraph. A real number $r\in (1,2)$ is called a Turán exponent if there exists a bipartite graph $H$ such that $\mathrm{ex}(n,H)=\Theta (n^r)$ . A long-standing conjecture of Erdős and Simonovits states that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q\gt p$ .

In this paper, we show that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q \gt p^{2}$ . Our result also addresses a conjecture of Janzer [18].