  In this paper we prove quantitative regularity results for stationary and
minimizing extrinsic biharmonic maps. As an application, we determine sharp,
dimension independent $L^p$ bounds for $\nabla^k f$ that do not require a small
energy hypothesis. In particular, every minimizing biharmonic map is in
$W^{4,p}$ for all $1\le p<5/4$. Further, for minimizing biharmonic maps from
$\Omega \subset \mathbb{R}^5$, we determine a uniform bound on the number of
singular points in a compact set. Finally, using dimension reduction arguments,
we extend these results to minimizing and stationary biharmonic maps into
special targets.
