On the back of a shampoo bottle in my friend’s shower, “damage level” is translated as “dégâts de niveau.” I find this odd, isn’t it backwards? Shouldn’t it be “niveau de dégâts”?
You are right. In general damage level should more translate to niveau de dégâts.
However, I have found a shampoo label showing both the English and French forms you refer too, and in that case, despite the Spanish and Italian translations being reversed (level de daño, livello de danno), dégâts de niveau makes sense in French.
The full English phrase is more like "Damage level: #3" while the French one can be read Dégâts de niveau 3. In that case niveau 3 is fine and idiomatic.
I don't know if these damage levels are standardized.
In France, we talk more about the agressivité of a shampoo, with a shampooing extra doux being a very low damage level one followed by a shampooing doux up to a shampooing très agressif but the latter won't be labelled like this.
After reading @PapaPoule's comment, I searched a little more and found an advertising that reads:
First, a salon will diagnose your hair’s ‘damage level’ (graded 1 to 3)
On the other hand, a reply on Amazon gives a range from 1 to 5.
Whatever the highest level is, this is about the hair quality. I don't think there is such hair grading used in France and dégâts might not be the best word to use. Both dégâts de niveau 3 and niveau de dégâts 3 sound odd when referring to hairs, that's more for a car or a house when talking with your insurance company. Perhaps qualité or dégradation would be closer to the idea.
Instead of a single range, we generally talk more about the kind of disorder the hair might suffer and say shampooing pour cheveux normaux, secs, très secs, gras, très gras, cassants...
You're right that the way to say “damage level” in French is “niveau de dégâts” (assuming a literal translation for “damage” and “level”).
“Dégâts de niveau 3” works, however. It depends how you complete the sentence fragment. If you complete it as “the level of damage is 3”, then it's “le niveau de dégâts est 3”. But you could have expressed it in English as “this product causes damage at level 3”, which can be phrased in French as “Ce produit cause des dégâts de niveau 3”. It wouldn't be a literal translation of the English, but it would be grammatically correct and semantically acceptable. I doubt that this is what the manufacturer meant, though. In both English and French, this formulation makes it more apparent that this is a negative aspect of the product. Using a noun phrase “damage level”/“niveau de dégâts” makes the negativity of “damage” a bit less prominent.
The expression “niveau de dégâts” is very uncommon (even in Canada) so I'm sure that the literal translation is not the correct one. I think that jlliagre is right that we'd use agressivité. French-speaking countries don't seem to put numbers on agressivité, but if we did, we wouldn't write ”niveau d'agressivité“, we'd just write “agressivité”, even with a numerical value, even in Canada.