Seventy-five percent of melanoma undergo progressive change over a period of six months to as long as many years, during which time the malignancy develops, but may not spread (metastasize) to other sites in the body.
I can't say for sure if 4 weeks is a long period, but I'll suggest your mum to try to make another appointment with other dermatologist.
A biopsy is typically an outpatient procedure, performed in the doctor's office with only a local anesthesia.
I don't like the fact that the lesion was bleeding.
Sometimes it can indicate more aggressive and advanced melanoma.
So, please, try to find another dermatologist who can make biopsy earlier.