The Electric Cage Match - Metro Square and possibly the Juggernaut DM - Watkyn's Harbor are two of the events that can give you trouble.
Electric Cage Match - Metro Square
This is where you're inside a green horizontally-lined area. If you exit out of it, your Grace Period (a sort of timer) starts to run down. The longer you stay out of the cage, the more your Grace Period goes down. If you exit the cage when your Grace Period expires, you'll slowly start taking damage. You can stop this effect for a little bit by using the Shield (press right on the D-Pad). The cage will move every now and then, so I recommend looking on your radar for a red spot. That's where the Electric Cage is. Get there before your Grace Period expires!
So for this type of event, I'd avise something like the Death Warrant or Kamikaze vehicles, as you need to be fast to survive this. The objective is to stay inside the cage for as long as you possibly can while killing off the enemies. Fortunately, there's a lot of Health Pickups on this map. Just be sure to be careful and not to stay outside of the cage. As soon as the cage moves, throw everything you've got at your opponents while you race there. If you get there before everyone else, just shoot at them from a distance while you wait, or grab some more pickups. Hopefully you'll get out of this event using these tips without too much trouble.
Juggernaut DM - Watkyn's Harbor
As explained by Calypso, there's this truck with a container in the back called Juggernaut. It is named as such because it has an insane amount of health. You need to kill him and the others to beat this event. Every few minutes, he will spawn a new enemy, which will open his doors = more damage when shooting there. There's also a point where you can shoot him in the front to do the same amount of damage. You need to hunt down this truck as fast as you can. If you destroy it before it spawns any enemies, you'll get the "Birth Control" trophy.
I'd recommend a vehicle that has high armor, such as Sweet Tooth or Outlaw. You'll stand a better chance against the Juggernaut. Just drive around, stocking up on weapons and then when you have enough, go find the Juggernaut. Shoot at it as much as you can. Hopefully you'll kill it before it spawns a wave of enemies. If you start to get low on health, get away from it, and find some Health Pickups and a few more weapons in case if you need more firepower.
When the Juggernaut is destroyed, it reverts back to the regular Death Match. Simply take out every car one by one, and you'll be finished in this hell of an event.
The Brothers Grimm
This boss fight is the final event for Sweet Tooth. It's easy, actually. Just stock up on weapons, shooting at them whilst doing so. There's a few Health Pickups, so be sure to grab them when needed.
Shoot one of them. Either kill off Slayer or Hammerhead, doesn't matter which one. Once the meter on one of them is depleted. Then the next phase of the battle will begin.
The steel shell will cover the other truck. You can't damage it because of that. You need to stay underneath the truck until you get Sweet Tooth's assistant to jump under it. Now you need to drive around, stocking up on weapons as you possibly can while the C4 is being planted. Once the C4 is at 100%, some dialogue will play. Once the Detonator is selected at the bottom of the screen, press R2 to use it. This will blow off the armor off of the truck.
Now you can get back to shooting at the truck. Using the weapons you stocked up on, it'll be an easy task. EMP's do work on the trucks so you can shoot them constantly. After killing The Brothers Grimm, the trophy will pop.