No, it is NOT normal, and YES you should have it looked at by a doctor
very quickly. While you don't mention anything about your age, history of sun exposure, family history, or any other factors, any bleeding
mole should be examined because it could possibly be cancerous. The mole mnemonic usually is ABCDE, where each letter stands for something to be concerned about with regards to moles.
A- Asymmetry. The mole is not regular on all sides
B- Border irregularity.
C- Color variation. (within the mole itself)
D- Diameter greater than 6mm. (roughly the size of a pencil eraser on the back of a normal pencil)
E- Evolving. (changing over time)
Your mole is in a special category: skin lesions that bleed spontaneously. These are quite often biopsied, which can be done in the office and will let you know just exactly what is going on. This is a quick, simple, and safe procedure that provides much needed information to you and your doctor. The important thing to remember is that a bleeding mole is not something that you should ignore. More than all of the other symptoms, spontaneous bleeding is your mole trying to tell you to talk to a doctor immediately.