Let $G$ be either the Grigorchuk $2$-group or one of the Gupta-Sidki $p$-groups. We give new upper bounds for the diameters of the quotients of $G$ by its level stabilisers, as well as other natural sequences of finite-index normal subgroups. Our bounds are independent of the generating set, and are polylogarithmic functions of the group order, with explicit degree. Our proofs utilize a version of the profinite Solovay-Kitaev procedure, the branch structure of $G$, and in certain cases, results on the lower central series of $G$.