Let $n$ and $g$ be positive integers, with $4\le g\le n$. What is the greatest number of edges in a graph on $n$ vertices if it contains no cycles of length less than $g$? This question from extremal graph theory has been studied by many researchers, but the answer is known for only very few infinite families of pairs $(n,g)$. The best general lower bound for the maximum number of edges comes from a family of algebraically defined graphs introduced by Lazebnik, Ustimenko and Woldar, usually denoted by $CD(k,q)$, where $k\ge 2$ is an integer, and $q$ is a prime power. It is known that for any $q$, the length of the shortest cycle (called the girth) in these graphs is at least $k+4$. It was conjectured that the girth of $CD(k,q)$ is $k+4$ for $k$ even, and $k+5$ for $k$ odd. The conjecture is wide open, and it was confirmed only for a few infinite families of pairs of the parameters $k$ and $q$. In this talk, we will present new infinite families of pairs $(k,q)$ for which the conjecture is true.