  We study integral curvature conditions for a Riemannian metric $g$ on $S^4$
that quantify the best bilipschitz constant between $(S^4,g)$ and the standard
metric on $S^4$. Our results show that the best bilipschitz constant is
controlled by the $L^2$-norm of the Weyl tensor and the $L^1$-norm of the
$Q$-curvature, under the conditions that those quantities are sufficiently
small, $g$ has a positive Yamabe constant and the $Q$-curvature is
mean-positive. The proof of the result is achieved in two steps. Firstly, we
construct a quasiconformal map between two conformally related metrics in a
positive Yamabe class. Secondly, we apply the Ricci flow to establish the
bilipschitz equivalence from such a conformal class to the standard conformal
class on $S^4$.
