It most certainly should not mean that. Newspapers is a countable noun in English, meaning that when it is in the singular, it must be followed by an article, in this case "a."
In Turkish, general direct objects do not take the plural suffix, unlike what is done in English. Kadın gezeteler okur is just not natural in Turkish. This means The woman reads newspapers and The woman reads a newspaper are both valid translations.
We go into more details in the notes as well, so check them out if you have the time. I hope that helped! :)