I am looking for existing constructions of vertex-symmetric graphs on $n$ nodes that have a girth at least $g$ and are dense, i.e., have at least $n^{1 + \epsilon}$ edges, where $\epsilon>0$ may depend on $g$: in particular, if $g = O(1)$, then $\epsilon$ must be a constant bounded away from $0$ (w.r.t. $n$).
I am aware of Cayley graphs, which are vertex-transitive, but I did not find any high-girth constructions that give dense graphs.
If you want an explicit construction (that does not involve randomness), you can use classical constructions of Ramanujan graphs, such as the one from this paper of Morgenstern.
Given an odd prime power $q$ and an even integer $d$, Theorem 4.13 in that paper gives you a Cayley graph on $n\sim q^{3d}/2$ vertices, with degree $q+1$, and girth $g$ at least (roughly) $2d$. So the number of edges is (roughly) $n^{1+3/2g}$.
Consider a Cayley graph on $\mathbb{Z}/n\mathbb{Z}$ with generators in a set $A$. In particular connect $x$ to $y$ is $x-y\in A$; notice that a short cycle in this graph corresponds to a linear relation among the elements of $A$. To have no cycle of length at most $g$ one only needs to avoid equations of length at most $g$ and a random set of size $n^{c_g}$ for appropriate $c_g$ suffices. For the best possible results on $c_g$ there is the literature of $B_h$-sets.
