  A graph $G$ is {$k$-crossing-critical} if $cr(G)\ge k$, but $cr(G\setminus
e)<k$ for each edge $e\in E(G)$, where $cr(G)$ is the crossing number of $G$.
It is known that for any $k$-crossing-critical graph $G$, $cr(G)\le 2.5k+16$
holds, and in particular, if $\delta(G)\ge 4$, then $cr(G)\le 2k+35$ holds,
where $\delta(G)$ is the minimum degree of $G$. In this paper, we improve these
upper bounds to $2.5k +2.5$ and $2k+8$ respectively. In particular, for any
$k$-crossing-critical graph $G$ with $n$ vertices, if $\delta(G)\ge 5$, then
$cr(G)\le 2k-\sqrt k/2n+35/6$ holds.
