I agree it is tough to see fine details in the small video, though.
First thing I see before you even start is to shorten those reins! They are WAY too long. When you pick up on the reins to stop, you want your reins about 4 inches in front of your saddle horn (length will vary slightly based on how long your arms are and how long your horse's neck is). You've got a fairly short-necked horse so you can go short.
Here's what my reins look like when I am just sitting and relaxing on my horse Red.
You have a very cute horse though! I do agree with that.
Alot of times, before you start your barrel run, it can be to your benefit to circle your horse before you start your run. This does a couple things
1)makes sure you horse is on the correct lead
2)makes sure your horse is soft and listening to you, and is focused.
Here's an example of circling before a run, with my horse Red, who I am just training for barrels. This was one of the first few times we loped the pattern. But I start him with circles to make sure he's picked up his correct lead and that's he focused and ready (he's normally quite ADHD so he certainly needs it). Not his best practice by any means, but you can see the circling beforehand. 2012-09-09 Saltys Red Rock barrel practice 01 - YouTube
On your first barrel turn, I'd give your horse a bigger pocket. He took the turn rather tight, which slowed him down, and made him lose his impulsion. Especially when you're first starting out of the barrels, you can go ahead and make your pocket size generous. As you add speed, they will naturally suck in closer to the barrel.
And you've already said you lean forward too much, which is true!
Especially on that first barrel turn, you are quite thrown forward in the saddle. This is where it would help to hold onto the saddle horn to keep yourself out of your horse's way. I will hold onto the horn with my outside hand, and then wedge my elbow into my outside hip. This reminds me to keep my weight centered and slightly back in the saddle. You also want to keep your weight slightly moreso in your outside stirrup. It is simple physics: An object going in a circle will go faster with the weight to the outside. So really focus on sitting back in the saddle.
As for the second barrel, spooks happen. I'll be trying to season Red this coming year and he usually boogers at quite a few things. But again, make that pocket larger. Don't turn the barrel so tight; so lose your speed by doing that.
Now one thing you did do right (that alot of people don't), is you went directly from the 2nd to 3rd barrel in a straight line. Perfect! You never want to arch or drift from barrel to barrel. Again, it's science: The fastest and shortest distance between two points is a straight line.
And then your 3rd barrel pocket .... was actually too big this time.
But I guess I'd rather see a pocket like that, versus a small one, because the too small pocket can cause you to hit barrels. You want to enter the pocket at about 4 feet from the barrel. I like to keep my distance the same all the way around the barrel, but some horses may have a different style than that, so you've just got to kind of feel how they are going to turn the quickest.
Overall, not bad!