You can buy shampoos that are specifically designed to thicken hair, such as this one: http://poiseandpurpose.com/blog/tlshair_sidebar/index.php?affid=370210&c1=TSI&c2=444 (although I think this is a tablet, so you might want to look up some shampoos)
Also, there are some hair-cutting tricks that can make your hair look thicker. Try some shampoos and talk to your hairdresser before going to the doctor. It might solve the problem and your mum will be more likely to approve of these.
If it's still not working, or your hairdresser might say that there is a problem, then go to your doctor. Your mum should be more open to the idea as you've already tried other things. If she's still doesn't want you to go, is there anyone else who could take you? Your dad or an aunt or someone? If not, and you feel okay with it, you could go by yourself. You can make the appointment yourself. The only difficult part would be how to get there and get back. I know that when you are over 16 there is a doctor-patient confidentiality which means that the doctor will not tell your parents you have been to see him/her, but I'm not sure if it's the same for under-16s. Good luck :)