That's the intereseting thing about riding , we are on an animal, with a mind of it's own. And when it comes to "steering", it isn't a machine that is literally made to go a certain direction. Rather, we use the reins and our legs and all to ask the horse to go in the direction we want. The horse has the choice to go along with our request, or ignore it. But, as you know, we must get obedience to our requests, so , yes, we must ask stronger.
Ask, tell, demand. First we kind of ask or suggest a turn, then we say "we are going right", then we say, "go right, now!". If you allow the horse to make those decisions himself, then he will make more and more decisions himslef and that could be frightening because you are on the back of a large animal that has no real connection to you. You then become a helpless passenger.
So, the long and the short of that, is that you DO have to be more assertive. It's good that you have a concern about not wanting to make the horse even harder in his mouth by hauling on the rein. But, unfortunately, you have to get him to respond and you will have to do whatever it takes.
YOu can either smack him, and I would , if he were just walking off in a direction that was in complete disobedience to my clear command. I would give him a smart little smack on the bum and when he jiumped, which he will, I'd reaffirm my command with the reins and get him mvoing forward again. Once you lose forward momentum it's harder to steer a horse. So sometimes, just get forward in any direction and then pick up the rein in the direction you want to go. Sometimes, if you get into a kind of tug of war, yoiu have to break things out of the deadlock.
In fact, when it comes to pulling on the rein, if I pick up a rein, say to go right, and the hrose is like a dead weight on the end, I will lift the rein higher and put a matching pressure to the horse and ONE OUNCE MORE. If the horse pulls on the rein, I will meet his pull but not exactly. If I meet it, I keep us in the place of a stalemate. You must not stop in the stalemate. So, meet his pressure and one ounce more.
The trick to not making the horse a hard mouthed horse is that when he does finally give to your pressure, you reward him with a bit release of pressure. If you do this a lot, and are very fair and consistent, you will build in greater willingness on the horse's part to respond sooner and lighter to your rein pressure. But you have to be spot on about giving a release AFTER he gives. HOres must give first!
On a schoool horse it is a real challenge and unlikely that you can really retrain him. So, you must becomes more assertive. STronger and use the crop if necessary. Watch how the trainer does it, and mimic her assertiveness. Don't be scared of angering the horse. He'll get over it.