  The extremal number $\mathrm{ex}(n,F)$ of a graph $F$ is the maximum number
of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number
$r \in [1,2]$ is realisable if there exists a graph $F$ with $\mathrm{ex}(n ,
F) = \Theta(n^r)$. Several decades ago, Erd\H{o}s and Simonovits conjectured
that every rational number in $[1,2]$ is realisable. Despite decades of effort,
the only known realisable numbers are $0,1, \frac{7}{5}, 2$, and the numbers of
the form $1+\frac{1}{m}$, $2-\frac{1}{m}$, $2-\frac{2}{m}$ for integers $m \geq
1$. In particular, it is not even known whether the set of all realisable
numbers contains a single limit point other than two numbers $1$ and $2$.
  In this paper, we make progress on the conjecture of Erd\H{o}s and
Simonovits. First, we show that $2 - \frac{a}{b}$ is realisable for any
integers $a,b \geq 1$ with $b>a$ and $b \equiv \pm 1 ~({\rm mod}\:a)$. This
includes all previously known ones, and gives infinitely many limit points
$2-\frac{1}{m}$ in the set of all realisable numbers as a consequence.
  Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart
from being interesting on its own, we show that, somewhat surprisingly, this
subdivision conjecture in fact implies that every rational number between 1 and
2 is realisable.
