Is it normal to have bleeding gums when brushing your teeth?

Healthy gingiva should not bleed. You appear to have gingivitis. See a dentist for an evaluation.

Dr Malerman agreed

If you practice good oral hygiene than no bleeding, but make sure you brush with a soft toothbrush and start brushing on your gums first than down onto teeth. See a real dentist for treatment, don't treat yourself until your dentist gives you the proper methods.

Gingivitis, inflammatory infection of gums, often presents first as bleeding gums, sometimes exacerbated by periodic hormonal swings. Most common cause is failure to remove bacteria from between teeth where t'brush cannot reach. Caught early this is reversible. See your Dentist ASAP.