Mainly because of cultural discrimination, lack of Christian focus on conversion, and lack of incentives. The Ottoman Empire was very tolerant of its Christian subjects as long as they paid the jizya prayer tax and contributed some of their young sons to the Janissaries. The disparate Christian states of Europe had neither the inclination or political resources to be able to integrate a population of Muslims. But since you ask about prisoner conversion, let’s consider the cultural differences first.

From the time of the Hellenistic period starting around 300BC, Asia Minor and the Near East has been a very culturally diverse place. Hundreds of cultures and dozens of civilizations rose and fell in the area prior to the arrival of the Greeks, but after the establishment of the Successor kingdoms (remnants of Alexander’s empire), a melding of Greek and “Asian” cultures began. This relative openness to new ideas added to the never before experienced ease of transport between places as far away as India gave enormous areas access to never before seen or heard ideas , technology, religions, trade goods, etc. This trend continued under Rome, the Byzantines, and eventually to the Ottomans. All in all, the East was a very educated and cosmopolitan place, with the Ottomans in particular being very good at recognizing that if a person converts, his skills could be of use later. Even up until 1685, Europeans had no issue with expelling anyone of a different religion, even if the expelled people had valuable skills. With the Edict of Fontainebleau, Louis IVX caused almost half a million Protestants out of France, losing untold intellectual potential and almost destroying the French economy in the process. With this cosmopolitanism also comes an understanding of race, that’s important later. Let’s contrast that with Christian Europe. Also under the control of Rome for a very long time (and Eastern merchants were a common site in places like Rome), but for the most part the “Asian” cultures that are widespread in the Near East and Asia Minor are seen only in trade hubs like Constantinople, Rome, Carthage, and port cities in Southern France like Massilia. Eastern cultures which were intermixed in the East as a matter of course were seen as “feminine” and degrading to many in the West.

Now let’s add in the race factor from earlier. That same cosmopolitan aspect of the East is the West’s Achilles heel (you could argue even today). While a Caucasian male would not be an oddity in Istanbul, an Arabic male would be an oddity (or worse if they were African) in most of Christian Europe. While a Caucasian prisoner might accept or be forced into Islam, they could have an actual life. Unfortunately for most Muslim prisoners there was not substantial incentive to actually convert since most of time Christians were more keen on ransom for nobles (or execution for levies). This is not necessarily because of racism per se, simply a lack of understanding due to a more homogeneous environment and natural disinclination to accept someone who doesn’t look like you (that’s not being racist, every civilization that meets another has a nearly identical knee-jerk reaction. The Romans thought the Asiatic Huns to be disgusting and the Greeks called Persians animals on more than a few occasions).

Let’s brush over the conversion process. First of all, the idea of organized Christian conversion prior to the New World is mainly a myth. Monks and friars did travel to Asia and the Near East starting around the 13th century, but before that conversion was mainly focused in regions with active Crusades (Levant, East Prussia, northern Britain) and it was usually forced. For Islam it was the mandate from the start, and let’s also remember that Islam is much younger (potentially making Muslim prisoners more fanatic, but don’t quote me on that). The Quran states no person should be forced into Islam, but it is the duty of Muslims to spread the word of God. Forced conversions have happened plenty in Muslim history as well (some of those Christian prisoners could have just been forced into Islam).

So that’s a lot to break down but if you as a Christian have a choice of converting to Islam, potentially rising within the bureaucracy, and being accepted as an actual person vs. being Muslim who converts to Christianity but being shunned and outcast, without much in the way of rights, and perpetually standing apart from your neighbors by virtue of your race, I know which position I’d much rather be in.

Edit: that’s disjointed and my formatting is trash but I wrote this on my phone and can’t fix it. Hopefully the point gets through.