  We investigate the Hilbert transform and the maximal operator along a class
of variable non-flat polynomial curves $(P(t),u(x)t)$ with measurable $u(x)$,
and prove uniform $L^p$ estimates for $1<p<\infty$. In particular, via the
change of variable, these uniform estimates are equal to the ones for the
curves $(P(v(x)t),t)$ with measurable $v(x)$. To obtain the desired bound, we
make full use of time-frequency techniques and establish a crucial
$\epsilon$-improving estimate for some special separate sets.
