In Engelking's General topology, in the exercises section, there is Ju. M. Smirnov's characterization of normal spaces:

A $T_1$ space is normal iff the following properties hold (both):

1. Every closed $G_\delta$ set is zero-set;
2. for every $F$ closed set and $G$ open set, such that $F$ is in $G$, there exist $M$ closed $G_\delta$ set, such that $F$ is in $M$ and $M$ is in $G$.

This equivalency is not hard to prove. Then there is written that, neither of the properties by itself imply the normality of $X$. For 2, maybe for example one may use the co-finite topology on a countable set, because any closed set is also $G_\delta$ (Also, for that example one may use Niemetzki plane by the same argument).

But I can't find an example of $T_1$ space, which has the property 1 and is not normal. Thanks for any help.