Realizability of rational exponents by a single bipartite graph

Let $r\in(1,2)$ be rational. For a graph $F$, write $\operatorname{ex}(n,F)$ for the maximum number of edges in an $n$-vertex graph containing no copy of $F$. Realizability of rational exponents. For every rational number $r\in(1,2)$, there exists a bipartite graph $F_r$ such that

$$\operatorname{ex}(n,F_r)=\Theta(n^r).$$

Bukh and Conlon proved the analogous assertion with a finite family of graphs in place of one graph; this conjecture asks whether every rational exponent can be realized by a single bipartite graph. The source also records the stronger Erdős–Simonovits question requiring an asymptotic formula $cn^r+o(n^r)$, but that stronger question is not the present claim.