In reading a paper, I came across an affirmation

"a graph of girth $g$ and $q$ vertices has at most $q^{1+(O(1)/g)}$ edges"

In a previous question I asked in this site about it, I was reffered to a theorem by Bondy and Simonovits, that states:

$ex(q,C_{2k}) < (k - 1 + o(1)) q^{1+1/k}$

This is a bound that does the job, assuming a fixed girth.

However, in my case, I have a girth that grows with $n$ (a variable independent of $q$).

So, now I ask a more generic question: **what kind of bounds are known, to limit a graph's number of edges, given its girth and its number of vertices?**