I was born in California and automatically became a US citizen. I obtained the second nationality from my parents. Now I live in Turkey and the government offered me Turkish citizenship without asking me to renounce any previous ones. Are there any negative consquences for a US citizen in this scenario?
The US does not generally restrict its citizens from having other nationalities. This doesn't depend on the number of other nationalities.
The same caveats that apply to dual nationality also apply, however. Taking another nationality is one of the statutory "potentially expatriating acts," but one cannot be deprived of US citizenship involuntarily, so citizenship is retained unless the act is done with the intention of relinquishing citizenship. The US generally presumes that no such intention exists.
You do not state what your other country of current citizenship is. That country might have a law that would cause you to lose its citizenship if you naturalize in Turkey.
Triple nationality is definitely possible.
My children have US citizenship (through their mother and their place of birth), UK citizenship (through me) and NZ citizenship (through naturalization).