Oh, good question.

So, taxation wasn’t intended to encourage conversion; it was actually intended to provide a legal status for the people who paid it as non-Muslims. However, it frequently had that impact because the tax rate levied on Muslims is specified in the Qur’an, whereas the one levied on non-Muslims was not, so it could be raised in times of economic hardship (or to encourage conversion — just because it wasn’t intended to be used that way doesn’t mean it didn’t happen, although I don’t recall offhand that this was the case among the Ottoman sultans).

Anyway, the short answer is this: among adult converts circumcision tended to be more of a “nice to have” than an “absolute.” In most Muslim societies, men are either circumcised at birth, or at set ages (6 or 7 seems to be the most common). An adult male who converted might do so as a sign of devotion, but it wasn’t strictly enforced religiously. Socially would be another case — as the bathhouse was a common place for business and political wheeling and dealing, exposure (pun intended) was always a possibility.

For a child the procedure would be done at a mosque or a shrine — sometimes they would have a dedicated individual on call, but often it was a barber-surgeon (in Arabic/Turkish/Persian the term was jerrah, and these men also performed basic surgeries, lanced boils, set broken bones, etc). Especially for an adult male it would be more likely to have been a jerrah.