Learn Turkish: Onun önsezisinin doğru olduğunu biliyordum.

This sentence states a known fact rather than a subjective impression. Because of the verb bilmek (to know), we use a factual noun clause with the -dık suffix on the verb olmak. If it were an impression, you would drop the noun clause entirely and use our current lesson's grammar: Onun önsezisi doğru gibi geldi (His hunch seemed right).

Can you break down the stack of suffixes on önsezisinin?

The root is önsezi (hunch or premonition). We first add the third-person possessive -si to make önsezisi (his hunch). Then, because his hunch is the subject of the nested noun clause, it takes the genitive suffix -nin, linking it directly to the verbal noun olduğunu.

What exactly are the suffixes on olduğunu doing?

It starts with olmak (to be). The -dık suffix (mutated to -duğ) turns the verb into a noun meaning 'the fact that it is'. The first -u is the possessive suffix tying back to the genitive önsezisinin (his hunch's being). Finally, the buffer -n- and the accusative -u mark the entire clause as the direct object of biliyordum.

Is the word Onun required here since the possessive suffix already shows whose hunch it is?