[slopbop]

Normal is actually a stronger claim than "contains any finite string as a substring". That normal numbers contain any finite string as a substring is a straightforward consequence of the infinite monkey theorem: https://en.wikipedia.org/wiki/Infinite_monkey_theorem

To see that the converse does not always hold, you could take something like the Champernowne constant https://en.m.wikipedia.org/wiki/Champernowne_constant and pad it with 9s between each integer, ie

.192939495969798999109911991299...

so that you still contain every finite substring, but you have a >50% chance of a randomly selected digit being 9.

[erdos4d]

I think there is a subtlety here that makes this fail. Normal does not imply that any finite substring exists, just that the probability of such a string existing is uniformly distributed within the space of possible values. There isn't any guarantee that you will actually see such a string, though you almost surely will.