  Given a graph $H$ and a positive integer $n$, the {\it Tur\'an number}
$\ex(n,H)$ 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 {\it Tur\'an
exponent} if there exists a bipartite graph $H$ such that
$\ex(n,H)=\Theta(n^r)$. A long-standing conjecture of Erd\H{o}s and Simonovits
states that $1+\frac{p}{q}$ is a Tur\'an exponent for all positive integers $p$
and $q$ with $q> p$.
  In this paper, we build on recent developments on the conjecture to establish
a large family of new Tur\'an exponents. In particular, it follows from our
main result that $1+\frac{p}{q}$ is a Tur\'an exponent for all positive
integers $p$ and $q$ with $q> p^2$.
