"The government exists to protect the public."
Translation:Devlet halkı korumak için vardır.
I thought about it again and I think it's not actually wrong. In daily speech people tend to use the word 'devlet' even when they refer to the 'government'. In the U.S. the difference is more obvious but in Turkey there is one state and one government even if the state is more than the government. So people use the word devlet a lot when they talk about governmental stuff in a simplifying way.
So devlet may mean government in daily speech but hükümet would never mean state. And the actual difference becomes valid as your speech gets more formal or serious.
P.S. But I don't really understand Duo. It is sometimes in favor of informal language and sometimes not.
I can only partly agree to your argumentation. In this case it is not necessary to provide people with a questionable translation. Remember that this is a multiple choice question, so it is in the hand of the teacher to present learners the best and most common translation for "government". And this is clearly not "devlet". However, I see your point and I agree to what you say. I would probably support accepting both "state" and "government" as two alternative translations for "devlet" (at least in some contexts, not in all), but only if learners are translating from Turkish to English. Here, the task is different and it unfortunately provides an unnecessarily unprecise translation.
Ah I see so I guess the difference is that "halkı" means everyone in Turkey whereas "vatandaş" means all Turkish nationals. If I was a lawyer I guess I would say that the government has a duty to try to protect both the public in Turkey and Turkish citizens outside the country. But luckily I am not a lawyer.