We show that for every rational number $r \in (1,2)$ of the form $2 - a/b$, where $a, b \in \mathbb{N}^+$ satisfy $\lfloor b/a \rfloor^3 \le a \le b / (\lfloor b/a \rfloor +1) + 1$, there exists a graph $F_r$ such that the Turán number $\operatorname{ex}(n, F_r) = Θ(n^r)$. Our result in particular generates infinitely many new Turán exponents. As a byproduct, we formulate a framework that is taking shape in recent work on the Bukh--Conlon conjecture.