That suicidal streak of Katara's that Gran Gran didn't know about? Well Iroh really really really knew about his nephew's. In fact, Katara telling Iroh that Zuko wanted to turn himself in is what really made him believe her. Fortunately, Iroh has long experience managing such idiocy on his nephew's part. Not much success, but lots of experience.

Thankfully for both mine and Iroh's sanity, Ozai doesn't find out about the Avatar thing until after that regrettable incident with the blue spirit, and then there are certain reasons his daddy's message doesn't reach Zuko right away (can't say more, you can't make me!) but yeah, Ozai is an uncle/nephew love killer.