  We study disorder induced topological phase transitions in magnetically doped
(Bi, Sb)$_2$Te$_3$ thin films, by using large scale transport simulations of
the conductance through a disordered region coupled to reservoirs in the
quantum spin Hall regime. Besides the disorder strength, the rich phase diagram
also strongly depends on the magnetic exchange field, the Fermi level, and the
initial topological state in the undoped and clean limit of the films. In an
initially trivial system at non-zero exchange field, varying the disorder
strength can induce a sequence of transitions from a normal insulating, to a
quantum anomalous Hall, then a spin-Chern insulating, and finally an Anderson
insulating state. While for a system with topology initially, a similar
sequence, but only starting from the quantum anomalous Hall state, can be
induced. Varying the Fermi level we find a similarly rich phase diagram,
including transitions from the quantum anomalous Hall to the spin-Chern
insulating state via a state that behaves as a mixture of a quantum anomalous
Hall and a metallic state, akin to recent experimental reports.
