  We derive local estimates of positive solutions to the conformal
$Q$-curvature equation $$ (-\Delta)^m u = K(x) u^{\frac{n+2m}{n-2m}} ~~~~~~ in
~ \Omega \backslash \Lambda $$ near their singular set $\Lambda$, where $\Omega
\subset \mathbb{R}^n$ is an open set, $K(x)$ is a positive continuous function
on $\Omega$, $\Lambda$ is a closed subset of $\mathbb{R}^n$, $2 \leq m < n/2$
and $m$ is an integer. Under certain flatness conditions at critical points of
$K$ on $\Lambda$, we prove that $u(x) \leq C [{dist}(x, \Lambda)]^{-(n-2m)/2}$
when the upper Minkowski dimension of $\Lambda$ is less than $(n-2m)/2$.
