By using a sharp quartic curvature pinching for the mean curvature flow in $\mathbb{S}^{n+m}$, $m\ge2$, we improve the quadratic curvature conditions. Through a blow-up argument, we establish both a codimension and a cylindrical estimate, which show that in regions of high curvature, the submanifold quantitatively becomes codimension one. In these regions, the submanifold is shown to be weakly convex and moves by translation or evolves is a self-shrinker. Additionally, a decay estimate ensures that the rescaled flow converges smoothly to a totally geodesic limit as time tends to infinity, without the need for iteration methods or integral estimates. Our approach relies on the preservation of the quartic pinching condition along the flow and gradient estimates that control the mean curvature in regions of high curvature.