Riding out of the ring is one of the best things for a beginner to help improve their riding and is good for the horse. Riding on your own is also a very important skill and perfectly fine for beginners. It is something many of us who grew up riding did a ton of growing up. Many of us also fell off a lot. You bounce back more easily when you are younger.
But all things need balance with safety and instruction with safety. Although horses are not machines and crap can happen.
I would not avoid riding out nor should I avoid this particular horse but you do need more help in how to ride out and basic safety. Stirrups should be adjusted before heading out. Other riders with you should stand near you if you do have to adjust something. When riding out, you never trot away from another rider without them knowing you are and if you are on the horse staying behind. Expect them to want to GO with the leaving horse. When falling you want to tuck and roll. You may think you held you head but honestly that is impossible. And by trying, you probably tensed even more. I bet you hit your head but even if you didn’t. You can slosh your brain around without hitting you head and the results are similar. I’ve give myself a mild concussion when I didn’t even fall off the horse. So be careful. Let yourself heal. When you ride again, ask your instructor to work on emergency falling.
We did a photo shoot at my farm on how to practice falling for Practical Horseman. The riders practiced falling off a moving gator. The article is worth reading for all levels and with discussion with your trainer. http://practicalhorsemanmag.com/article/horseback-riding-falls-jim-wofford-27838