Hi,
No – this is not normal. It’s appalling practice actually.
My advice to you would be to contact CSCI (www.csci.org.uk) which is the regulatory body for all care homes in England and Wales. You can search and read all the homes’ inspection reports if you wish.
I would advise you to telephone them as opposed to sending an email, so that you can provide detailed information about your concerns. You can remain anonymous if you wish, although if you do provide details, you will receive feedback on what action they are going to take.
It is totally unacceptable to say that your mother can stay only if you do not question them. This amounts to nothing other than a threat which CSCI will take very seriously.