Dr said that breast cancer grows very fast in young, lactating women. said it could easily double in size in two months. possible?

Yes: Breast cancers in young women are usually high grade which means the cells are rapidly and actively dividing which can cause a rapid increase in tumor volume over a relatively short period of time. Most younger women have hormone receptor negative cancers, but if the hormone receptors happened to be positive - then lactational hormones would also pay a role in increasing the rate of proliferation.

Short answer is yes: Breast cancer in pregnancy or during lactation, especially in young women can be very aggressive. It is important to have a biopsy to confirm that the lump is, in fact, cancer and then seek appropriate treatment. Most lumps in young lactating women are fortunately not cancerous.