Board Thread:Questions and Answers/@comment-3186449-20130408114617/@comment-9059954-20130430231702

I think the idea was that the nuclear reaction that destroyed Krypton transformed every shattered bit of the planet into kryptonite. Therefore, the people of Krypton were generally dead in the same moment that kryptonite came into being.

It took a while for the ideas to settle out. In Supergirl's origin story (1959?) the chunk of Krypton holding Argo City was said to have become kryptonite, but in later retellings of the origin, it was said to have become a special form of kryptonite, called anti-kryptonite (IIRC), that affected only non-super-powered Kryptonians.