I can't see much on the 3rd barrel because it's tiny and far away, but the wide turns are about the only thing I noticed as well.
You should be following these arrow lines exactly.
Make sure you are looking at your axis points (on the ground) which are the points you want to cover with your horse. Make sure you follow all 4 axis points around the barrel you are completing fully, before you look up for the next barrel. Then make sure you head directly for your next axis point to start the pocket on your next barrel.
You are making an arc on your pattern, especially the 2nd to 3rd can be seen easily with the camera angle. Do straight lines. Not an arc.
Although looking back, I noticed on the first time you are just walking around the pattern, you simply neck rein him around the barrels. Do you do that at higher speeds? No. So don't do it at the walk. You should use your body cues in the exact same way you are going to do at all gaits. So if you need to two hand around the barrels at the trot, make sure you do it at the walk. Stay consistant.