  Let $p$ be an odd prime and $F_{\infty}$ a $p$-adic Lie extension of a number
field $F$ with Galois group $G$. Suppose that $G$ is a compact pro-$p$ $p$-adic
Lie group with no torsion and that it contains a closed normal subgroup $H$
such that $G/H\cong \mathbb{Z}_p$. Under various assumptions, we establish
asymptotic upper bounds for the growth of $p$-exponents of the class groups in
the said $p$-adic Lie extension. Our results generalize a previous result of
Lei, where he established such an estimate under the assumption that $H\cong
\mathbb{Z}_p$.
