We will discuss the mean curvature flow of $n$-dimensional submanifolds in $\mathbb{R}^{N}$ satisfying a pinching condition $|A|^2 < c|H|^2$ introduced by Andrews and Baker ('10). We will compare these flows to flows of hypersurfaces studied in the fundamental works of Huisken ('84) and Huisken-Sinestrari ('99, '09). Our primary goal is to discuss an estimate for these pinched flows which shows such flows must become nearly planar prior to singularity formation. At the end, we will discuss how to use pointwise derivative estimates to show the (necessarily codimension one) singularity models must be noncollapsed for suitable $c$.