In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field $\mathbb{F}_q(x)$ whose class groups have elements of a fixed odd order. More precisely, for 𝑞, a power of an odd prime, and 𝑔 a fixed odd positive integer $\geq 3$, we show that for every $\epsilon > 0$, there are $\gg q^{L\left(\frac{1}{2}+\frac{3}{2(g+1)}-\epsilon\right)}$ polynomials $f\in \mathbb{F}_q[x]$ with $\deg f=L$, for which the class group of the quadratic extension $\mathbb{F}_q(x,\sqrt{f})$ has an element of order 𝑔. This sharpens the previous lower bound $q^{L\left(\frac{1}{2}+\frac{1}{g}\right)}$ of Ram Murty. Our result is a function field analogue which is similar to a result of Soundararajan for number fields.