Teeth whitening should not normally cause your gums to bleed as long as you don't leave the teeth whitening chemicals on your teeth too long.
However if you do leave the teeth whitening products on your teeth too long the chemicals used in teeth whitening products can lead to irritation of your gums and inflammation of your gums.
That irritation and inflammation of your gums and bleeding of the gums is due to prolonged exposure of the chemicals used in the teeth whitening products.
If you do have teeth or gum bleeding from the teeth whitening products you should avoid leaving the teeth whitening products on your teeth for too long.
And if the bleeding continues from your gums you should consult your dentist about it.