I can't imagine NOT being supportive of any of my daughter's. I'm sorry you have to go through this. I had a self proclaimed 'hip mom' back in the day, yet she couldn't bring herself to tell me about sex or my periods even. I had to watch a video about them. A relative was the one who eventually got me to a Planned Parenthood to get an exam and BC.
When our oldest daughter started high school we decided she needed BC. She wasn't (and as far as we know, still isn't) sexually active. We know how fast the world is moving and how fast kids seem to be growing up these days. We want our daughter's to be aware that they have to be responsible for their bodies and what goes on with them. My husband is very involved with the kids, and he takes our oldest to get her BC shots every 3 months. His work is more flexible than mine, so he is elected to go. The nurses LOVE that he is involved. Not many parents, much less dad's, want to hear/see/know about their daughter's reproductive health.
When our other daughter starts high school, she too will be placed on BC, unless she asks for it earlier, or needs it earlier due to periods.
Your mom is resentful of the fact that you sent her out of the room. She should be ok with that to sone extent. It doesn't make her look like a bad parent at all. My husband and I have both left the rooms for medical exams for our oldest. (Him it's a no brainier, ha!) I volunteer to leave, it's her body and if she has questions or concerns to ask the Dr, then so be it.
Maybe it's because I am in my mid 30's or because I swore I'd never be like my mother, but I just can't imagine NOT ever helping or understanding my child's needs for something like this. I'm sorry she has decided to punish you this way.
I am not happy with the thought that my daughter could be having sex at 16, I would hope that she would wait a few more years, so she is a bit more mature, but I know my wants don't always matter to a 16 yr old. So I figure we better pick our battles, we need to educate her on her body, and go from there.