I remember seeing somewhere that in Battle Mode there's a control to quickly "turn on the spot", but can't find the details anywhere. What button(s) do you press?
According to this video:
You can perform a quick turn in Battle Mode by initiating a drift, then letting go of the gas button mid-drift.
But that's if you're already moving forward and need to turn around. If you need to quickly turn from a (near) standstill; hold the break and gas at the same time and steer in the desired direction. Your tires will squeal and your kart will turn on a dime. Release the brake to speed forward in whatever direction you are currently facing.