Well, nitrites do become nitrates, but that isn't the issue. The nitrates are not high because you had a problem with nitrites.
The issue is that the tank is being allowed to accumulate nitrates at too fast a level. More fish will make it worse, not better. And as the fish you already have grow .. and they seem to have a lot of growing to do... they will produce nitrates at much higher levels. When a fish goes from 4 inches to 8 inches, it produces 8 times as much waste.
So, even if you change your practice and start doing more and larger water changes, you will find that you still can't keep up. High nitrates with small fish that are still growing is a warning sign that things are heading in a bad direction.