Your back injury has lingered for 6+ months? You said you can't go to a doctor, but did you check with your doc after his 4-6 week timeframe passed and you weren't any closer to healing?
Dude, you need to be proactive about your recovery, not just be eager to get back to squatting three days a week. Get back to where you can do full range push-ups, pull-ups, bodyweight squats, and full range toe touches without pain, and then think about lifting.
Disbelief being the key word here, as apparently it was a band-aid solution that didn't "cure" anything. We should try not to do that again.
A 100% healthy body has zero discomfort and zero pain. At 16, that's where you should want to be. If you're feeling discomfort/pain, you're not fully healed, period. Tissue quality work (foam rolling, treating mobility drills as important as oxygen, etc.) should become part of your new daily routine. A handful of drills, 10-15 minutes a day everyday, to hit the whole body would be a start.
Also, consider some yoga. No joke. It'd be a decent combo of stretching and activity as long as you stick within your capabilities. Either find some beginners lessons online or see if you can find a local class to score a free trial.
Undertand that if you don't legitimately fix this 100%, it can be something that sticks with you as a problem for years. It's undoubtedly worth putting in the time to correct it now.