If you have a fracture, then yeah, rest is what you should be doing. Keep a good posture and let the bone heal. Then rebuild yourself.
I have bulging discs in between my L4 - L5 as well as L5 - S1. Much more minor in nature than yours.
Physio for a while which helped me strengthen my core a bit, but the thing that stopped 95% of the pain was losing weight.
I was carrying more than I should have and lost just 5ish kgs and the pain is gone for 95% of the time.
Where I struggle is building strength again. When I did my back, I was useless for 3 months and in that time, my strength just plummeted and I gained the weight. Once I dropped the weight, I still didn't have strength and when I would try and go back to the gym or do things some what strenuous, my back would be in pain again. Not to the point where I could feel it shooting down my leg, but it was there.
I knew that I couldn't handle high impact stuff. So I started running (which I'm enjoying) and yoga. Running gave me my confidence back. I had moments where mid run, my back would start to ache again, but I fought through it and it eventually went away. Posture is important.
Yoga was the best thing. It strengthened me just enough to the point where I'm back at the gym now (not as much as I like and feel like I'm embarrassing myself with the weights I'm doing). It also stretched me out, opened my joints up, made me more flexible, etc etc. I felt good again. And it wasn't the type of yoga where you would hold a pose, it was transitioning into a position, hold it for several breaths, then transitioning into another position. It really did get my heart rate up and felt like I was doing something. After some sessions, my muscles would hurt a bit the next day.
In short, start small and re-build. Get advice from several experts and get into a routine that is best for you. If you rush it, you're going to do more damage. Especially seeing as your bone is weak.
