  We consider knots whose diagrams have a high amount of twisting of multiple
strands. By encircling twists on multiple strands with unknotted curves, we
obtain a link called a generalized augmented link. Dehn filling this link gives
the original knot. We classify those generalized augmented links that are
Seifert fibered, and give a torus decomposition for those that are toroidal. In
particular, we find that each component of the torus decomposition is either
"trivial", in some sense, or homeomorphic to the complement of a generalized
augmented link. We show this structure persists under high Dehn filling, giving
results on the torus decomposition of knots with generalized twist regions and
a high amount of twisting. As an application, we give lower bounds on the
Gromov norms of these knot complements and of generalized augmented links.
