They conquered most of it.
The fact that the conversion rate on the Balkans during the Ottoman period remained pretty low has several explanations, but chiefly it was an economic issue. Christian subjects paid higher taxes than Muslims, plus they supplied conscripts for the empire’s elite janissary corps. If all the Balkan Christians has converted to Islam it would have created a huge deficit in the Ottoman state finances.
However, several Christian prominent families joined the Ottomans during their conquest and some subsequently converted to Islam as a way to maintain their rights and privileges. Furthermore, as the Ottomans secured their control over the peninsular, conversions were largelyade in order to access jobs in the state apparatus and civil service, positions which were exclusively reserved for Muslims. Those who converted though, largely retained their native tongue - Turkish being seen as a primarily administrative language.
Generally speaking, the Christian (and Jewish) minorities weren’t treated too badly during this period; there are even examples of Christians attempting to have their cases heard at sharia courts!
It was only with the onset of nationalism and the independence movements of the 19th century that an outright hostile dichotomy between the two faiths arose in the region. Islam was identified with Turkish suzerainty, which is why so many Muslims were forcibly expelled from Greece, Bulgaria and Serbia in the wake of their eventual independence. In Bulgaria for example, this persecution continued all the way to WW2 where the government attempted a violent forced conversion of the Pomak minority.