We consider Type I Ricci flows and obtain integral estimates for the curvature tensor valid up to, and including, the singular time. Our estimates partially extend to higher dimensions a curvature estimate recently shown to hold in dimension three by Kleiner and Lott (Acta Math 219(1):65–134, 2017). To do this we adapt the technique of quantitative stratification, introduced by Cheeger–Naber (Invent Math 191(2):321–339, 2013), to this setting.