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](https://mathoverflow.net/questions/243815/looking-for-source-max-num-of-edges-of-graph-with-given-number-of-vertices-and "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?**



Bounds for number of edges of a graph, given girth and number of vertices
The bound you quote is actually a little inefficient, because it is harder to find a cycle of fixed length $t$ than to find a cycle of length at most $t$. I recommend that you look at Section 4.1 of the excellent survey by Füredi and Simonovits: 


<https://arxiv.org/pdf/1306.5167.pdf>


In particular, Theorem 4.1 of that survey, saying that
$$\textrm{ex}(n, \{C\_3, C\_4, \dots, C\_{2k}\}) < \frac{1}{2} n^{1 + 1/k} + \frac{1}{2} n,$$
gives a fairly satisfactory answer to your question for odd girth.


