  We continue our investigation of the parameter space of families of
polynomial skew products. Assuming that the base polynomial has a Julia set not
totally disconnected and is neither a Chebyshev nor a power map, we prove that,
near any bifurcation parameter, one can find parameters where $k$ critical
points bifurcate \emph{independently}, with $k$ up to the dimension of the
parameter space. This is a striking difference with respect to the
one-dimensional case. The proof is based on a variant of the inclination lemma,
applied to the postcritical set at a Misiurewicz parameter. By means of an
analytical criterion for the non-vanishing of the self-intersections of the
bifurcation current, we deduce the equality of the supports of the bifurcation
current and the bifurcation measure for such families. Combined with results by
Dujardin and Taflin, this also implies that the support of the bifurcation
measure in these families has non-empty interior.As part of our proof we
construct, in these families, subfamilies of codimension 1 where the
bifurcation locus has non empty interior. This provides a new independent proof
of the existence of holomorphic families of arbitrarily large dimension whose
bifurcation locus has non empty interior. Finally, it shows that the Hausdorff
dimension of the support of the bifurcation measure is maximal at any point of
its support.
