Translation:There are no ships on the Turkish rivers.
The thing is, any number or another word denoting a quantiy or amount leaves the noun in the singular.
- sok hajó
- kevés hajó
- öt hajó
- 200 hajó,
The word "nincsenek" is not one of those words. It is not modifying the noun. It is actually the predicate of the sentence, and it means "there are no...".
And the predicate usually matches the subject in number. Those "number-like" words above usually stand in a modifying position, like an adjective in front of a noun. Such an adjective also remains singular:
piros hajók - red ships.
But a predicate does not follow that rule. If I make a predicate out of the adjective "piros", it immediately becomes plural, to match the plural subject:
A hajók pirosak. - The ships are red.
This is very similar to how "nincsenek" works. It is the predicate, it matches the subject in number.
Hajók nincsenek. - There are no ships. (Actually, this one sounds like "Ships do not exist.")
But the predicate can move around in a sentence:
Pirosak a hajók.
The rule still stands.
"Nincsenek" is actually a verb. But a special one, as it is only used in the third person. "Nincs" and "nincsen" are the singular forms, "nincsenek" is the plural form.
You can think of "nincs" and "nincsen" as variants of the same word. They are both third person singular. The "-en" ending used to be a third person singular ending, it is archaic now, but traces of it can still be found here and there in the language.
Here is my source, in Hungarian, for advanced learners: