The ship and crew Zuko has at the beginning of Avatar: the Last Airbender are actually Iroh’s. I believe Zuko was tossed out with nothing and forced to go around the world on foot but Iroh used some of his pull on the Fire Nation military to get Zuko a crew to make it a little easier for him.
I dunno, this just makes sense to me. Ozai doesn’t seem like he gave a single shit about Zuko. Definitely doesn’t seem like he’d give him resources to hunt down the Avatar.