I'm seeking "best possible" examples of a nonnormal quotient of a normal space, namely, a nonnormal but completely regular $T_{0}$ space $Y$ that is the quotient of a normal $T_{1}$-space $X$.
The more "concrete" and "geometric" the better!
Are there any such examples other than those constructed by the method from https://math.stackexchange.com/a/1682677/32337 (which uses ultrafilters) or https://math.stackexchange.com/a/1569463/32337 (which uses cardinality)?
Examples that avoid both cardinality and ultrafilters would be desireable.possible.
Related: Examples of a quotient of a normal topological space that is not normal?. Note that the quotient spaces in the accepted answer there are either not regular or not Hausdorff, so do not constitute a "best posssible" example.
It was shown by Hanai that a $T_1$ space is first-countable if and only if it is an open image of a metric space. See
I believe the result is also often credited to Ponomarev.
Since every open surjection is quotient, to answer your question will suffice to find any first-countable, completely regular, $T_1$ space which is not normal. The Niemytzki plane works.
This particular example can be made explicit by noticing that the Niemytzki plane $N$ is locally metrisable. Thus let $\mathcal{U}$ be any open cover of $N$ by metrisable open sets and let $X=\bigsqcup_{U\in\mathcal{U}} U$ be the disjoint union of the members of $\mathcal{U}$. The obvious mappings $\pi:X\rightarrow N$ is a continuous open surjection of a metrisable space onto a non-normal space.