Could veneers work to improve my teeth? (Photo)

Yes, veneers could likely create a substantial cosmetic improvement in your smile. However, based on the amount of crowding visible in your photo—particularly the upper canines—I would strongly consider orthodontic treatment first with either Invisalign or traditional braces.
In your case, the crowding appears significant enough that getting the teeth into better alignment first would most likely allow for a better and more natural-looking final result. You may even find that once the teeth are straightened, you are happy enough with the improvement that veneers are no longer necessary.
If you ultimately still want veneers after orthodontics, having the teeth in a better position generally gives the cosmetic dentist more ideal tooth proportions to work with and can reduce the amount of correction that needs to be accomplished with the veneers themselves.
That said, orthodontics is not always required before veneers. A moderate amount of crowding can often be visually corrected very effectively with carefully designed porcelain veneers alone. The important consideration is how severe the crowding is and how far the teeth would need to be altered in order to create the appearance of a straight smile.
With the degree of crowding I see in your photo, I believe orthodontics first would be the best option to consider if you are willing to go through the additional treatment time.
If braces or Invisalign simply are not an option for you, veneers could still potentially provide a significant improvement. I would just go into treatment understanding that the maximum aesthetic result would most likely be achieved by straightening the teeth first and then deciding whether veneers are still needed.

Hey, I think it is awesome that you want to improve your smile!
Veneers alone will likely NOT be enough!
I Highly Recommened getting braces First. Then Veneers AFTER braces.
Your Bite is not Ideal. Veneers alone will not work, and if you could get someone to place them, you like likely knock them off or break them with a bite like this.
Good Luck on your journey, it make take a couple years before you get the final result, but you will be happy you did!

Veneers are not recommended in your case. You have severely malaligned and crowded teeth. So, the ideal recommendation is orthodontic treatment, also known as braces or Invisalign treatment. That is for teeth alignment. Once your teeth are aligned, you can then go for veneers to improve your teeth and smile. But at this stage, veneers are not recommended.