  A stabilization operation is defined for codimension $2$ contact submanifolds
in $\dim \geq 5$ contact manifolds $(M, \xi)$. The definition is such that (1)
a given $(M, \xi)$ is overtwisted iff its standard transverse unknot is
stabilized and (2) transverse stabilization preserves the formal contact
isotopy class and intrinsic contact structure of a link. We prove that many
transverse links are non-simple.
