The main result is that the Jacobian of the Drinfeld modular curve X_1(n) associated to the prime n\in {F}_q[T] and the congruence subgroup \Gamma_1(n) of GL(2)) has connected reduction modulo the place n. This determines one of the bad Tamagawa factors in the L-function of the Jacobian. The proof rests on constructing a function field analogue of the Igusa curves which describe the bad reduction of elliptic modular curves, as well as on the resolution of cyclic quotient singularities on arithmetic surfaces.