2 weeks post hair transplant and newly transplanted hairs are breaking off at the root

You should see your doctor for a recheck.  Transplanted hairs usually slide out after several days to weeks.  They do not break off at the root.  Non-transplanted hair may fall out due to shock loss from surgery.  It is unclear what you are exactly describing so it may be best to have an examination.

What you are seeing is the normal shedding that occurs after a hair transplant and this is not a problem what-so-ever,  Most if not all of the transplanted hairs will shed in the next few weeks and then regrow in 5+ months

Hair goes through a cycle of growth. After hair transplant the hair shaft detaches from the follicle and comes to the surface and falls off. It will then start to grow in after a few months.