The Dowling geometry $Q_n(Γ)$, where $Γ$ is a finite group, is a matroid that generalizes the complete-graphic matroid $M(K_{n+1})$. We determine the maximum size of an $N$-free submatroid of $Q_n(Γ)$ for various choices of $N$, including subgeometries $Q_m(Γ')$, lines $U_{2,\ell}$, and graphic matroids $M(H)$. When the group $Γ$ is trivial and $N=M(K_t)$, this problem reduces to Turán's classical result in extremal graph theory. We show that when $Γ$ is nontrivial, a complex dependence on $Γ$ emerges, even when $N=M(K_4)$.