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! :)
Is gazete here an accusative or a nominative case? If it is accusative, it is supposed to be "the newspaper(s)", isn't it? 2 things here are confusing: 1) the fact that direct objects do not take on plural form, this has been explained by Alexin above, 2) the accusative case in Turkish and the definite article in English, how do they correlate, i do not understand (somehow "Ben et yerim"="I eat meat" and "Ben eti yerim"="I eat the meat", yet " Ben gazete yerim"="I read newspapers", why??)