  We show that the conjugacy problem in a wreath product $A \wr B$ is
uniform-$\mathsf{TC}^0$-Turing-reducible to the conjugacy problem in the
factors $A$ and $B$ and the power problem in $B$. If $B$ is torsion free, the
power problem for $B$ can be replaced by the slightly weaker cyclic submonoid
membership problem for $B$. Moreover, if $A$ is abelian, the cyclic subgroup
membership problem suffices, which itself is
uniform-$\mathsf{AC}^0$-many-one-reducible to the conjugacy problem in $A \wr
B$.
  Furthermore, under certain natural conditions, we give a uniform
$\mathsf{TC}^0$ Turing reduction from the power problem in $A \wr B$ to the
power problems of $A$ and $B$. Together with our first result, this yields a
uniform $\mathsf{TC}^0$ solution to the conjugacy problem in iterated wreath
products of abelian groups - and, by the Magnus embedding, also in free
solvable groups.
