Rational exponents near 3/2

Given a graph $H$, the extremal number $ex(n,H)$ is the maximum number of edges in an $n$-vertex graph not containing $H$ as a subgraph. The well-known rational exponents conjecture of Erd\H{o}s and Simonovits states that for any rational $\gamma \in (1,2)$ there exists a single bipartite graph $H$ satisfying $ex(n,H)=\mathrm{\Theta }(n^{\gamma })$. Among other results, the conjecture has been verified for all $\gamma =1+a\mathrm{/}b$, where $b>a^2$, by Jiang and Qiu and for all $\gamma =2-a\mathrm{/}b$, where $b>\max \{a,(a-1{)}^2\}$, by Conlon and Janzer. In this paper, we establish the rational exponents conjecture for many $\gamma$ near the center of the interval, namely, for all $\gamma =1+\frac{rt-1}{2rt+2r}$, where $r,t$ are natural numbers satisfying $t\ge 2$, $r\ge 2t+3$.