Car spins when steering is turned to an extreme?

When the wheel is turned all the way in one direction or the other the car begins to spin in that direction as if it had a life of its own, like it was caught in a whirlpool. This happens in reverse as well.

The power steering fluid was just replaced as well as the brake fluid and the transmission fluid and the oil and the spark plugs because I just bought the car used and have no knowledge of the previous owner’s maintenance history. Otherwise, the car is running great.

Also, there is a slight "squirm" in the steering wheel as I hit the extreme limit of where the steering wheel can turn to while backing up as well as going forward (when going at low speed like in a parking lot).