  We study high codimension mean curvature flow of a submanifold
$\mathcal{M}^n$ of dimension $n$ in Euclidean space $\mathbb{R}^{n+k}$ subject
to the quadratic curvature condition $ |A|^{2}\leq c_n |H|^{2}, c _n = \min\{
\frac{4}{3n} , \frac{1}{n-2}\}$. This condition extends the notion of
two-convexity for hypersurfaces to high codimension submanifolds. We analyse
singularity formation in the mean curvature flow of high codimension by
directly proving a pointwise gradient estimate. We then show that near a
singularity the surface is quantitatively cylindrical.
