  We prove that for the mean curvature flow of two-convex hypersurfaces the
intrinsic diameter stays uniformly controlled as one approaches the first
singular time. We also derive sharp $L^{n-1}$-estimates for the regularity
scale of the level set flow with two-convex initial data. Our proof relies on a
detailed analysis of cylindrical regions ($\varepsilon$-tubes) under mean
curvature flow. The results are new even in the most classical case of mean
convex surfaces evolving by mean curvature flow in $\mathbb{R}^3$.
