No infinite family exists. In fact all graphs with diameter $d$ and girth $2d+1$ have to be regular, and thus are <a href="https://en.wikipedia.org/wiki/Moore_graph">Moore graphs</a>. This was proved in
> R. Singleton, "There is no irregular Moore graph", Amer. Math. Monthly 75
(1968), 42–43

See also the texbook "Algebraic Graph Theory" by Godsil and Royle (p. 90).
It is unknown whether the Hoffman-Singleton graph is the largest Moore graph of girth 5 but there can be at most one more, of degree 57, whose existence is unknown.