  We prove an $\epsilon$-regularity result for the tracefree curvature of a
Willmore surface with bounded second fundamental form. For such a surface, we
obtain a pointwise control of the tracefree second fundamental form from a
small control of its $L^2$-norm.Several applications are investigated. Notably,
we derive a gap statement for surfaces of the aforementioned type. We further
apply our results to deduce regularity results for conformal minimal spacelike
immersions into the de Sitter space $S^{4,1}$.
