Does anyone have any good warmups/exercises that I could do to work on speed and tight turns to make the barrels tighter
Welcome OP!
It would also be helpful to know more about you and your horse, to make better recommendations.
How old is your horse?
How long have you had him/her?
How long as your horse been barrel racing?
Is your horse fully trained on barrels or still learning? (assuming your horse is still learning, based on the context of your post)
"Speed" and "tight turns" are things that usually come with time and with hauling. You want your horse broke, broke, broke before you start teaching them the pattern. Then as they gain confidence and understand what you want, you slowly increase the speed. Your turns should always be CORRECT no matter what speed you are going (slow or fast). It might fall apart a little bit here and there as you ask for more speed, and sometimes that's okay because you can always see where they are, what they need, and slow it back down again.
For working on turns, sometimes I will do "all rights" or "all lefts" with the pattern. Or sometimes just do that with two barrels, going over 360 degrees around each barrel. It seems to help teach them to keep engaged in the turn and learn to stay stood up to get all the way around.
And so it's not really "drill specific" but it's just about make sure they are always in correct position whenever you do go to turn a barrel, on whatever drill you are doing.