If my newborn has a bright red birthmark, will it fade away by itself?

They usually fade. Light colored birthmarks on the forehead usually fade. Larger birthmarks on the face and body may not. Strawberry birthmarks are not usually present at birth, but develop later. These usually resolve by about age 5 years, and sometimes leave a pale area of skin.
Most do with time. The answer depends on the birthmark. If it is flat and on the forehead or eyelids, it is called a nevus flammus and usually disappears by 2 years old. If raised it is a strawberry hemangioma, and may take up to 7-8 years to fade, but can be treated with medicine if on the face/neck, is very large or growing rapidly. Another type is a port wine stain. Your md should be able to tell which type.