We certainly can see galaxies that are younger than our own. And we can see galaxies as they appeared when they were much younger than our own galaxy is now.
Our galaxy is over 10 billion years old, though the thin disk formed a bit later. (By comparison, the Sun and Earth are less than 5 billion years old).
If there's a galaxy that formed, say, 4 billion years ago, and it's 1 billion light-years away, then we'll be able to see it as it was 1 billion years ago (when it was 3 billion years old).
A galaxy that formed at the same time but that's currently 6 billion light-years away would not be visible; we can see its birth if we wait 2 billion years.
More generally, if we see an object x light-years away, we're seeing it as it was x years ago. If it's less than x years old, we can't see it at all. If it's more than x years old, we're seeing it as it was when it was x years younger than it is now.
But I think (I'm not sure of this) that most galaxies formed at about the same time. Our own, for example, formed not long after the Big Bang. But even if all galaxies formed at the same time, if we look 10 billion light years into space, we can see galaxies as they were when they were much younger.
I'm oversimplifying a bit. Relativity tells us that simultaneity is a more complicated concept than we might think. It's not necessarily meaningful to say that something 6 billion light-years away happened 4 billion years ago. And the expansion of the Universe messes things up a bit as well.