You shouldn’t be fighting or hanging on for dear life. You should be relaxed, strong, and in control of your body’s movements.
Of course you’ll get tensed up a bit in the corners to stay upright and not flop around in the seat, but you shouldn’t be struggling with it at all.
Don’t worry about bruising. I think it’s safe to say most of us have some bruises from the seat after a hard day’s work.
Watch some videos of the European guys driving and you’ll see how they keep their posture in the kart.
You always have to keep in mind that you are roughly 50% of the weight of the entire vehicle. Any shifting of your body will greatly affect the kart so it’s best to keep an upright and stable posture. Unless the kart is handling poorly and you need to get your lean on a bit to get it to rotate.