Hi, Gabrielle. That is a very good question. Technically, "I've noticed ~~" is equivalent to" ~~ 한 것을 눈치챘다." / "~~한 것을 깨달았다." / "~한 것을 알아차리다".
1. I noticed that I've improved a lot. : 내 실력이 많이 늘었다는 것을 깨달았다.
2. She noticed that he was passing notes in class. 그녀는 그가 수업시간에 노트를 돌리고 있다는 것을 눈치챘다.
3. I noticed that you were doing that at the time. 나는 너가 그 당시에 그것을 하고 있다는 것을 눈치챘어.
4. He noticed that she was staring at him. 그는 그녀가 그를 쳐다보고 있다는 것을 알아차렸다.
However, we don't really say "I've noticed" that often in Korea. For example, rather than saying "I've noticed", I would say "I saw he was passing notes in class (나는 그가 노트를 돌리는 것을 봤다)" or "It seems like I've improved a lot. (나 실력이 많이 는 것 같아.)"
I hope this helps you.