There are a number of reasons why gums may bleed when brushing. The bristles of your toothbrush may be too stiff, in which case, invest in a toothbrush with softer bristles. You can also run your toothbrush under warm water to soften the bristles.
You may also be using too much force when you brush. Be gentle yet thorough when brushing.
Lastly, sensitive and/or bleeding gums could be a sign of a more serious problem such as gum disease. It’s recommended to talk to your dentist to determine what should be done.