Learn Turkish: Bakırda hasar olduğunu fark ettin, değil mi? - You noticed that there is damage on the copper, didn't you?

In Turkish, the confirmation tag değil mi? (literally "isn't it?") is invariable. It doesn't change to match the tense, person, or verb of the main sentence. Whether you mean "didn't you?", "aren't they?", or "won't we?", you always just use değil mi? at the end.

How does olduğunu work here to mean "that there is"?

To say you noticed that something is the case, Turkish uses a noun clause with the -dık suffix. For facts involving existence ("there is damage"), we use olmak (to be). It becomes olduk, takes the 3rd person possessive -u (referring to the damage), a buffer -n-, and the accusative case -u because this entire clause is the direct object of what you noticed.

Is fark ettin a single verb? Why is it written as two words?

It is a compound verb made from the noun fark (difference/realization) and the helper verb etmek (to do/make). These are written as separate words unless the noun part changes its spelling (like dropping or doubling a letter). Since fark does not change, fark etmek stays as two words.

Could I say "bakırın hasarı" instead of "bakırda hasar"?