GIVE AN EXAMPLE :
X:normal
Y:normal
BUT X X Y : not normal
normal is not productive
The classic example (given in Kelley's General Topology, and attributed by him to Dieudonné and Morse, independently) uses ordinal spaces. Let be the set of all ordinal numbers less than the first uncountable ordinal and let , each with the order topology. Then and are both normal, but is not. (Let and . Then A and B are closed and disjoint in but do not have disjoint open neighbourhoods. The proof is not that easy. You would do best to look it up in Kelley's book.)
Another example is a Sorgenfrey plane .
Even though a Sorgenfrey line is normal, the Sorgenfrey plane is not normal.
Let . As shown by the above link, no disjoint open sets can separate the disjoint closed sets and in . Intuitively, unions of the open rectangles containing and unions of the open rectangles containing in the above link somehow overlaps all the time. A rigorous proof needs more elaboration.
In contrast, and are disjoint closed sets in a Sorgenfrey line , which can be separated by disjoint open sets in (they are both clopen sets).