  We study the $L^2$-type contraction property of large perturbations around
shock waves of scalar viscous conservation laws with strictly convex fluxes in
one space dimension. The contraction holds up to a shift, and it is measured by
a weighted related entropy, for which we choose an appropriate entropy
associated with the strictly convex flux. In particular, we handle shocks with
small amplitude. This result improves the recent article [18] of the author and
Vasseur on $L^2$-contraction property of shocks to scalar viscous conservation
laws with a special flux, that is almost the Burgers flux.
