Oh gosh, I had that when I started biking a lot. Mine was a mountain bike and that is the worst handlebar ever. A different handlebar made it a lot better, but we did a lot of tinkering, longer stem, adjusting the height and tilt of the handlebar, and eventually I got a bike that fit me better.
No, this is not something that will resolve as you get stronger at riding. In fact it will just get worse and like the others said, there's a possibility of permanent damage. I hope you find a solution that works for you!
2009 Trek 7.2FX WSD, brooks Champion Flyer S, commuter bike