Whatever you do, act like an adult and have and adult conversation with your mom.No whining or pouting.
I can understand your mom being worried, I still worry about my 33 yr old daughter, its just the nature of the beast, we are moms afterall.
Have her go to a horse show with you. Have your watch lessons being given by the barn/trainer you want to go to. Let her observe in her own time that horses can be very fun to be around and most are not dangerous. Encourage her to talk to you about her feelings and then you can talk to her about yours.
Explain to her that driving is more dangerous, after all, a drunk driver can hit you, even though you are being careful. Riding a bike is dangerous, taking a walk is dangerous with predators around. At least riding a horse is done in a safe closed corral , with a helmet on and with an instructor helping/training. Do you have brothers who are in sports? Boys have been paralyazed in football, but they still play.
Be honest and have a really good one on one chat with her about this, talk over your feelings and hers and see if you can come to an agreement to at least try it for a bit and see how it goes. Tell her you want her to come watch you ride, include her.