  A longstanding conjecture of Erd\H{o}s and Simonovits states that for every
rational $r$ between $1$ and $2$ there is a graph $H$ such that the largest
number of edges in an $H$-free graph on $n$ vertices is $\Theta(n^r)$.
Answering a question raised by Jiang, Jiang and Ma, we show that the conjecture
holds for all rationals of the form $2 - a/b$ with $b$ sufficiently large in
terms of $a$.
