When you're doing some slow work, trot or lope up to the barrel and stop (maybe a length before the barrel). Then make your turn, exaggerating your pocket.
I had a horse that tried to run over the first barrel, and here's how I fixed it. We did a lot of slow work, including the above. Sometimes, I would counter-arc him before we turned the barrel because he was a little heavy. I also set up a cone about 30 ft from the barrel (for the first barrel, I put the cone on the third barrel side straight up the fence, does that make sense?). I would turn the barrel with a big pocket and then lope three big, relaxed circles around the cone. On the last circle I would stop when I was facing the second barrel and let him stand for a minute, then ask him to lope off on the left lead. We did the same thing for all three barrels.
I learned this last exercise at a clinic, and it was extremely helpful. It helped tremendously with finishing turns, but also taught him to not cut off the back side of the pocket.
Good luck, and let us know how things are going.
Learning never stops