Exponents of class groups of real quadratic function fields (II)

Let $g$ be an even positive integer. We show that there are $\gg q^{l/g}/l^2$ polynomials $D\in\mathbb F_q[t]$with $\deg(D)\le l$ such that the ideal class group of the real quadratic extensions $\mathbb F_q(t,\sqrt D)$ have an element of order $g$.