The claim is false it would imply that normal spaces are countably paracompact and hence that normality of $X$ would imply normality of $X\times[0,1]$. The latter is not the case, see Mary Ellen Rudin, A normal space $X$ for which $X\times I$ is not normal, Fundamenta Mathematicae, 73 (1971/72), 179-186.
To show that the property in the claim implies countable paracompactness we use Theorem 5.2.1 in Engelking's General Topology. Let $\{U_n:n\in\omega\}$ be an increasing open cover; we need to find open $O_n$ such that $\operatorname{cl}O_n\subseteq U_n$ for all $n$ and $\bigcup_nO_n=X$. Take an open cover $\{V_m:m\in\omega\}$ a cover as in the claim; hence for every $m$ an $n$ such that $\operatorname{cl}V_m\subseteq U_n$. Now define $O_n=\bigcup\{V_m:m\le n$ and $\operatorname{cl}V_m\subseteq U_n\}$; then $\operatorname{cl}O_n\subseteq U_n$ for all $n$ and the $O_n$ form a cover (if $x\in V_m$ and $\operatorname{cl}V_m\subseteq U_n$ then $x\in O_n$).