Can you shift your nose cartilage by bending it too much?

Generally speaking, no, you're not likely to affect the shape of your nose from what you're describing. It's MUCH more likely that the shape changes you're noticing are related to continued growth and development of your nose as your adult genes are being expressed. This is why we typically recommend waiting to undergo rhinoplasty until that development has largely stabilized. The best way to check it out would be to get an opinion from a surgeon trained in both otolaryngology (ear, NOSE, and throat) and in facial plastic surgery.All the best,
--DCP
Pearson Facial Plastic Surgery®

It is not possible for you to cause a permanent change in the shape of your nose by pushing it as you describe. To change the shape of cartilage as you describe you would have top break it which would be very painful.  Please don't worry.  It is likely as you have thought for yourself, that the changes are due to your nose not having finished growing.

Hi and welcome to our forum!
The cartilages of the nose are quite flexible. The upper 1/3 of the nose is bone and the lower 2/3 of cartilage, able to tolerate the traumas of daily life. I don't think you can alter nasal shape by bending the cartilage.. If done with extreme force, the cartilage could theoretically be fractured or the overlying perichondrium damaged, but there would be significant associated pain, swelling and overlying redness following the episode.
Most likely continued growth is the cause of the increased crookedness. 
Best wishes...