"I read newspapers and drink water."
Translation:Ben gazete okurum ve su içerim.
"Gazete" is a general direct object: you're not referring to a specific newspaper, but to newspapers in general. Because of that, you don't need a plural ending: "gazete okurum" can be used to mean either "I read a newspaper" or "I read newspapers."
If you do add a plural ending, you also need the accusative ending, so it becomes "gazeteleri okurum." But that's a better translation of "I read the newspapers." So basically:
I read a newspaper. -> Ben (bir) gazete okurum.
I read newspapers. -> Ben gazete okurum.
I read the newspaper. -> Ben gazeteyi okurum.
I read the newspapers. -> Ben gazeteleri okurum.
The accusative "tips and notes" should help: https://www.duolingo.com/skill/tr/Accusative
For some reason, Turkish plurals need to be in the accusative case: you can skip the accusative ending with a singular noun, if it's a general direct object, but if your noun has a plural ending, the accusative ending is obligatory. (I don't know why that is; a native speaker might.)