For a straight turn, I always use leg and rein on the same side when neck reining, like you said. Left rein + left leg = right turn. It's not until later in his finishing that you would need to worry about using both legs in a turn to create bend. You've got the basic idea but you need to have the progression a bit different. Put leg and neck rein on at the exact same time, then
direct rein to reinforce your cue almost immediately if he doesn't respond. When he turns, then release all pressure. If done correctly, it doesn't normally take too much time for them to catch on to the basic idea of it. I can have a horse that is neck reining decent after only a couple of rides where I was actively working on it.
The best way to get them used to it quickly is to not really do set times where you spend only a bit of your ride working on neck reining. It should be a constant work in progress. When you decide you want him to start learning, then you need to work on it every single second that you're riding him. Start expecting him to neck rein when you're riding on the trail, when you're in the arena, when you're riding down the road, when you ride from the arena to the trailer, etc. The more consistently you ask for it, the more consistently he'll pick it up and the quicker he'll be consistent with the cues.
Once I start really teaching a horse to neck rein, I will start riding them with 1 hand and will only ever use the other to reinforce a cue if he is a bit sluggish to the initial cue or if he seems confused.
Another problem most folks run into when they start trying to teach a horse to neck rein is that they pull the rein across the horse's neck to the point that it cues the bit. The problem that creates is that if you are pulling the left rein across the horse's neck trying to get him to turn right but the left rein is tight enough that it's pulling on the left side of the bit, then the horse is getting conflicting signals. The rein says "turn right" but the pull on the bit says "turn left". What normally happens is either the horse turns the wrong way or he does one of those ugly turns where he turns right with his nose pointed around to the left.
Here is a pretty good video about teaching neck reining. Not the very
best example I've ever seen, but it works.