  Let $T_1$, $T_2$ be two singular integral operators with nonsmooth kernels
introduced by Duong and McIntosh. In this paper, by establishing certain
bi-sublinear sparse domination, the authors obtain some quantitative bounds on
$L^p(\mathbb{R}^n,\,w)$ with $p\in(1,\,\infty)$ and $w\in A_p(\mathbb{R}^n)$
for the composite operator $T_1T_2$. Some weighted weak type endpoint estimates
are also given.
