Work on collection. When you are loping circles in the beginning, her head is in the air, back is hollow, and she is not reaching underneath himself with her hind legs. It makes her look stiff and choppy. I myself have a hard time understanding how to ask my horse to travel collected, so I took a reining lesson on Saturday. Wow, what a big help! Then I could feel what my horse is supposed to feel like when he is reaching underneath himself with his back legs. Main point --> Go take a lesson or two from someone who can help you get your horse collected at all gaits, and what it should feel like.
Ditto to above poster. Ask for bend, bend, bend! And keep your horse's hip inward. When you are walking the pattern in the beginning, you let her walk through the turn while her body is straight. When you make a turn (even at a walk) there should be nice bend in her body.
At 2:56 when you backed her up before the barrel, she turned her hip out. Don't let her do that. Hip needs to stay INWARD so that she can properly engage it during the turn.
When you go to lope through at 4:06 you are on the wrong lead. On a finished horse, it doesn't matter because they know their job. On a horse who is learning? You need to help them and cue them for that correct lead to start the pattern.
Overall, the run that started at 4:06 isn't bad at all. The only thing I would change about your pattern is to head STRAIGHT for your pocket on the 3rd barrel. You are actually heading to the right of the pocket, and then need to straighten before you turn. That costs time to do it that way.
There might be some stiffness there. I'm not entirely pleased with how your horse used her hind end in the pattern. It looks choppy.
How old is this horse?
Has she ever had a lameness eval or checked by a chiro?
It could also be that she's just not using her body effectively because you aren't asking her to collect and bend. Or she could be stiff.
Use your legs. Not your hands. Look at your approach to the barrel. Your horse's nose is out. How can she properly complete a turn when you are setting her up like this?
Nose needs to be in. Ribcage out. Hip in. Like this: