Good points from Minimor. Let's start with your equipment and your horse. First of all, how old is he and how often have you been driving him? A young horse or one who is out of shape (and especially one who is both!) will have a hard time balancing through a corner and may need to be shown how to manage the cart. Secondly, do you have a picture of your horse harnessed up and preferably making a turn? An unbalanced cart, one with shafts that are too long or especially shafts that are too narrow will make it very difficult for your horse. If he can't physically move his hip and ribcage over, he can't bend and will be forced to sidestep. If the shaft tip is poking him in the shoulder he may also lose forward momentum and once he's dropped to a walk again he gets confident enough to step over into it.
Assuming your equipment is not causing the problem and he's physically fit enough to do what you're asking, go back to basics. The goal in bending is to have the horse's body form an arc from his nose to his tail along the line of travel, coming up through the poll and stepping under himself with his inside hind leg. His nose tipping to the inside is only a side effect of real bending which takes place in the ribcage. When you ask the horse to give with his nose you want him to soften his jaw and come up into the bridle. When he does this it should cause a chain reaction- he softens his jaw, gives at the poll, raises the root of his neck, frees his shoulder, lifts his ribcage and lets his hind leg swing further forward. When this happens his energy should be collected and recycled through his body. Somewhere in this process it sounds like you've taken a wrong turn and his energy is dumping out the front. The most likely reason for this to happen is that you're asking for him to give his nose and then expecting him to follow it but not pushing him up into the bridle and helping him find the power to do so. He has to be driven from back to front! FIRST the forward energy, THEN the give in the jaw, THEN the physical turn.
Make sure you go back to long-lines for these lessons and don't proceed to cart work until he really understands them. When you do hitch, do everything at a walk and quit when he does it right! Come back and work on trot another day when he's had time to rest and think about it. If he can't do it at a walk, he sure as heck won't be able to do it going faster.
The other part of this equation is your hands. As Minimor said, there are different cues for asking a horse to sidestep through a turn versus bending through one. To get a bend you want to half-halt on the inside rein to ask the horse to give his jaw, set up the bend, then
allow the turn with your outside rein. The outside rein must continue to support the horse and dictate the size of the turn- it can't just be loose while you pull on the inside rein. Think of it as a supporting leg against the horse's ribcage, showing him how far over he can move. If you're only pulling on the inside rein the horse will either stop in confusion when the shafts start pushing into him or he'll do the only other thing he can figure out, which is sidestepping over following the pressure. If your horse is not used to feeling contact on both reins through a turn he will also likely stop in confusion- ask him to go forward into the soft contact and praise him for doing so then ask gently for a turn again. It helps if you can do this in a large area where there is no fenceline to run into! If you don't have a big field, introduce the concept in gentle serpentines going down the road then when he's going well you can have him make a big slow sweeping turn in a driveway and do serpentines all the way back home. They often get the idea quickly this way as it's fun for them.
Leia
Edited to add: As an example, here's a picture of my pair. The silver buckskin on the off side is a three year old colt who has only been hitched a few times and doesn't yet have the strength to keep trotting for long or stay in balance through his turns without help. He was tired and cranky by the time this photo was taken and while he's tipping his nose in the direction of the turn, he has no impulsion and is heavy on the forehand. By himself he would be breaking to a walk. The chestnut on the near side is my 11 year old Preliminary level horse and even when out of shape, sore from the previous day and shackled to his uncooperative partner
he is up in the bridle, bending through his whole body and stepping up under himself. The difference is impulsion!
If I were driving the colt by himself at that moment I would have been taking up strong elastic contact, driving him up into the bridle then allowing him to make the turn with a lot of support from the outside rein and praising him as he went forward.