He isn't really using himself at all. What I mean by that, is that he's turning but he's not using his hind end or adding any drive into the turn.
At the first, you bow off wide (He's also very stiff in his shoulders) and on the backside he immediately crossfires in the hind. He needs to stay in that lead the whole way around the barrel and not switch until he has left it. That tells you right away he can't drive off his inside hock because he isn't loping on his inside hock - He's loping on his outside which throws him off.
Approaching the second you duck and weave a little bit. That's just anticipating and very common, but you see him do that same thing - Switching his leads, shouldering, and that is not conducive to getting a turn with him driving. That is why he is not snapping, he has no idea what to do with his body.
Watch these videos and try to achieve this with him. Exaggerate the inside bend and leg, and get him broke through his nose, ribcage, and hindquarters. That will be the start to it.
This is the next step, keeping them on a circle and encouraging that lead to stay in position.
If you cannot get him to keep his leads and position his body, he will never get the drive you are looking for.
He is displaying typical green horse behavior, so don't be discouraged. Once you get him a little more broke he will learn to shape himself. Just remember it is muscle memory, you have to work him on the pattern and shape him every day perfectly in order for him to understand. If you let him get away with it every now and then in practice, he will continue to throw his hip around like he did in that run.
This is a video of my little mare for good measure, she is my lesson horse now, but watch in every photo where her hip is - Up underneath of her. She was young in these videos so she still makes mistakes just like yours does, but you can see her naturally trying to correct that because she knows she has to.
Now this is her "sister" (They aren't full sisters but they come from remarkably similar breeding). She is much farther along than my mare is. Watch how this mare turns - You won't find a snappier or quicker turner than this mare, never. She turns like she is supposed to - Shoulder up, hip in, inside leg up under her and she DRIVES.
There are more advanced drills to help as well, but try this first. And if you think your horse is past that, tell me, I am more than happy to post some other advice.