  We prove that for the mean curvature flow of closed embedded hypersurfaces,
the intrinsic diameter stays uniformly bounded as the flow approaches the first
singular time, provided all singularities are of neck or conical type. In
particular, assuming Ilmanen's multiplicity one conjecture and no cylinder
conjecture, we conclude that in the two-dimensional case, the diameter always
stays bounded. We also obtain sharp $L^{n-1}$ bound for the curvature. The key
ingredients for our proof are the Lojasiewicz inequalities by Colding-Minicozzi
and Chodosh-Schulze, and the solution of the mean-convex neighbourhood
conjecture by Choi, Haslhofer, Hershkovits and White. Our results improve the
prior results by Gianniotis-Haslhofer, where diameter and curvature control has
been obtained under the more restrictive assumption that the flow is globally
two-convex.
