"Screening causes breast cancer." I take that as Peter Gøtzsche's brand of humor. Needless to say screening mammography doesn't actually turn normal cells into cancerous ones. What Gøtzsche means is that screening by its nature detects abnormalities and cancers which are not destined to ever harm or kill. That is, screening causes overdiagnosis. And overdiagnosis almost always leads to overtreatment (doctors will opt to do something rather than nothing, because we currently can't distinguish between early stage cancers that will become life threatening and those that won't). Because the cancer being treated isn't destined to be aggressive, overtreatment harms the patient (unnecessary lumpectomy or mastectomy, radiotherapy with its suite of harms, psychological harms, all the well known nasty effects of chemotherapy, etc.).
[image source: https://www.youtube.com/watch?v=8A9xuLmUHcQ]