If I understand correctly, the reason for "turning Turk" -- converting to Islam -- was that the Ottoman Empire had agreements with the Hapsburgs and other European states that they would not shelter traitors or rebels from each other. However, there was an exception: the Ottomans would not hand a fellow Muslim over to a Christian state for punishment. So, if a Christian rebel fled into Ottoman territory, and then converted to Islam, they would be safe.

Obviously there would be a lot of problems with this course, from adult circumcision to learning the Turkish language. Also, this was an irrevocable decision; under Ottoman law, backsliding from Islam was punishable by death. So, a number of the rebels of 1848-9 escaped into Ottoman territory, but then refused conversion, and so were handed back to face harsh punishment, from years of imprisonment to death.

However, some chose to convert; and as you say, sometimes the rebels were prominent nobles or experienced generals or administrators, men with talents the Ottoman Empire could make use of. Bem in particular was an extremely competent commander. He's not well known in the English-speaking world today, but Stonewall Jackson was a great admirer of Bem's campaign in Transylvania, and apparently cited it as an inspiration for his own campaign in the Shenandoah Valley fourteen years later.

As to why not Italy, that's easy -- Italy was within the Habsburg reach. Most Italian states were either friendly to Austria, or under Austrian domination, or at least too wary of Austria to risk sheltering high-profile fugitives. Much the same applies to the German states, and of course Czarist Russia would hardly be friendly to rebels. In theory, if a rebel could somehow reach Second Republic France, they might find shelter there -- but good luck getting there across hundreds of kilometers of hostile territory.

So, realistically, it was the Ottomans or nothing; and once across the Danube, the rebel would be faced with a snap decision to convert or be sent back.

Two other notes. First, while Bem had the title of "Pasha", which is often translated as "Governor", he wasn't the chief administrator of Aleppo. That was a different Pasha. (IIUC, all governors were Pashas, but not all Pashas were governors.) However, during the political troubles there, Bem seems to have unilaterally seized a lot of authority. So, one still sometimes sees references claiming that he was "the governor" of Aleppo.

Second, at the time of Bem's death there were rumors that he had died, not of malaria as was officially claimed, but rather from poison. Certainly that sort of thing was not unknown in the late Ottoman Empire. And Bem -- an upstart foreign convert who was exceeding his authority to take charge of suppressing a popular rebellion -- would have been an entirely plausible target. On the other hand, Bem was already 55 years old, had been repeatedly and seriously wounded in his final campaigns, and by all accounts had not completely recovered his health. And he was probably under a lot of stress from the conversion, a new country, language, and diet, and so forth, not to mention fighting yet another battle. So, him succumbing to malaria in the unfamiliar heat of Aleppo is also quite plausible. We'll probably never know for sure.