The claim is false it would imply that normal spaces are countably paracompact and this 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][1].
To show countable paracompactness we use Theorem 5.2.1 in Engelking's *General Topology*. Let $\{U_n:n\in\omega\}$ be an increasing open cover and $\{V_m:m\in\omega\}$ a cover as in the claim; we then have for every $m$ an $n$ such that $\operatorname{cl}V_m\subseteq U_n$. Define $O_n=\bigcup\{V_m:m\le n:\operatorname{cl}V_m\subseteq U_n\}$; then $\operatorname{cl}O_n\subseteq U_n$ for all $n$ and the $O_n$ form a cover. 


  [1]: https://matwbn.icm.edu.pl/ksiazki/fm/fm73/fm73121.pdf