Hi Charlie - sometimes it's the bit that's making the horse strong! Some horses have very soft mouths and prefer a snaffle, or if that's too light some prefer a bit with more poll action than mouth action (a Dutch gag, for example) or even a bitless bridle that uses poll, nose and chin pressure rather than mouth pressure like a hackamore.
Another thing to think about is when and why your horse starts pulling down and running. Videos are very helpful for this. Does your Haflinger start grabbing the bit and running when you use the reins to try to stop/slow down? Maybe he does it when he gets frustrated with something.
Regardless of what's causing him to do it, it's important to look past how to control his behavior and start thinking about how to fix it so you can slowly work towards a softer bit and ride for both of you. This includes not only looking at what he's doing but also at what you're doing! Unfortunately, a lot of the time we do things without realizing that can accidentally cause problems for the both of you! Again, this is where a video would come in handy. I'm certain that with some trial and error and figuring out the cause of his leaning and running you'll get him to where he's nice and soft 100% of the time. Best of luck!
You've probably heard it and I know it's irritating but it's usually not about the bit. Leaning horses are hard to ride. Try being as light in hand as you can and if he wants to put the head down ask him to move forward. Otherwise, waterford is great. Sometimes horses would fight against the stronger bit so maybe you could try a snaffle :)