Is it normal to have filler injected in the nose dissolve after 3 weeks? (Photos)

filler always goes down after a few weeks and touchups are often needed.  I suggest you go back to your physician to see what is best for the result you are looking to achieve.  Best, Dr. Emer.

Filler makes the nose look bigger. In your case, by filling out the radix area, just above the bump, it makes the nose look longer. It's just almost never ever a good idea.
If you want to consider a rhinoplasty, almost certainly nothing special would have to be done with your radix area. Your bump would be lowered, the tip elevated a touch, and the entire nose moved back closer to your face a little, too. 
Rhinoplasty is an exquisitely difficult operation to get right, and you should only have surgery if you are able to make yourself very confident in your surgeon's skills. The changes I described above require advanced techniques, requiring skill that most plastic surgeons don't possess with expertise. For most noses, it's much better to not have surgery than to have inexpertly-performed surgery. Read my essay on how to stay out of trouble while selecting a rhinoplasty surgeon.
Your nose is also a perfect example of why computer imaging is mandatory in rhinoplasty. You need to know exactly what the surgeon is planning to accomplish -- what features he thinks he can change, and by how much he thinks he can change them. I always email morphs to my online rhinoplasty consultations. When you see the surgeon's goals in the morphs, you'll know whether he has an eye for an attractive nose, and whether he shares your opinion of what constitutes an attractive nose. You'll also know whether the changes he proposes are large enough to be meaningful to you, and whether he understands your wishes enough to address all of your priorities. How would you ever get that figured out without the morphs?! 
But remember, you're not the surgeon for his skills with the computer. The doctor must then show you his before and after photos to prove that he can actually accomplish what he draws on the computer.