Show that a closed subspace of normal space is normal
Let $X$ be normal space and $S$ a closed subspace of $X$. Now let $A,B$ be disjoint closed subsets of $S$. By the subspace topology we have that $A=A' \cap S$ and $B = B' \cap S$ for $A',B'$ closed in $X$. Since $X$ is normal there exists open sets $U$ and $V$ such that $A' \subset U$ and $B' \subset V$ for which $U \cap V = \emptyset$.
Does this imply that $S$ is normal or do I need to consider something more?
Since you did not use the fact that $S$ is closed and since not every subset of a normal space is normal, that cannot possibly be correct. The error appears when you write that “there exists open sets $U$ and $V$ such that $A′\subset U$ and $B′\subset V$ for which $U\cap V=\emptyset$.” Why do such sets exist? You have no reason to suupse that $A'$ and $B'$ are disjoint.
Note that, since $S$ is closed, $A$ and $B$ are closed in $X$ too. And they are disjoint. So…