  We test the cosmic censorship conjecture for a class of polarized AdS black
branes (holes) in the Einstein-Maxwell theory at large number of dimensions
$D$. We first derive a new set of effective equations describing the dynamics
of the polarized black branes (holes) to leading order in the $1/D$ expansion.
In the case of black branes, we construct `mushroom-type' static solutions from
the effective equations, where a spherical horizon is connected with an
asymptotic planar horizon through a `neck' which is locally black-string shape.
We argue that this neck part (of black string) cannot be pinched off
dynamically from the perspective of thermodynamical stability. In the case of
black holes, we show that the equatorial plane on the spherical horizon cannot
be sufficiently squashed unless the specific heat is positive. We also discuss
that the solutions are stable against linear perturbation, agreeing with the
thermodynamical argument. These results suggest that Gregory-Laflamme type
instability does not occur at the neck, in favor of the cosmic censorship.
