My next dive.
If you are trained as you should be, you should be able to dive independently as soon as you are certified. That's what the cert means. If you you don't feel capable of being an independent diver upon certification, then your instructor did you a disservice by certifying you.
With all due respect, I felt completely capable of making a dive independently with my recently certified buddy "in the conditions I trained in".
We trained in low vis, dark and cold fresh water quarries and lakes, from shore. I felt 100 percent comfortable making quarry/lake dives from shore in bad vis, cold nasty water, no problem. When you are in a quarry, you simply can't go very far wrong on navigation... and when the quarry is only 63 feet deep it is kind of hard to mess up by diving deeper than the recreational limits.
I did not feel comfortable diving from a boat, in salt water, in a wide open ocean where navigation is actually a challenge.
We signed up for, and took, our AOW courses not so much for the information that they taught and the certification cards but rather for the fact that our first five dives in the ocean would be with an instructor working through not only the coursework but all the other things that freshwater trained divers have not seen.
I think it worked out pretty well... when we were done with our AOW we were ready to just tell the folks on the boat "hey, we are going to go do our own thing" and they were good with that (especially because our own thing was to simply stay within 100 feet or so of the mooring and examine the macro animal life and corals closely).
OW Certification, in my case, made me ready for conditions that I had trained in... but AOW training in the ocean added several necessary tools to my diving skills.
DISCLAIMER: I am a NOOB, so keep that in mind when listening to anything I say... I do not offer advice on diving because I don't know what I am talking about... I only answer questions about what I DO know about (which is how I was trained and what I have seen on the limited number of dives that I have been on).