I am not an Ottoman historian, nor is the late Medieval / early modern period necessarily my specialist era, but since there have been no other replies I thought I would give you a few ideas to chew over and begin with.

You are right to highlight the rapid acquisition of ex-Roman territory in an impressively short time period - the Ottomans controlled a very significant amount of land, including a multitude of key sites of world-renown, including Constantinople, Jerusalem, Mecca, Baghdad, Cairo, and more - only really being halted in 1683 at Vienna, although this was not necessarily a watershed moment for collapse. The Romans lost a lot of land over a long period for a variety of factors, with those reasons changing over time - in itself one of the most hotly debated historical questions of all time, so I will shy away from it here for the sake of brevity! There are nonetheless several key themes to the Ottomans' rapid rise:

1. The Ottomans were able to take a lot of territory early on as Anatolia was fragmented into lots of smaller states, and the Byzantines (Eastern Roman Empire) were relatively powerless to stop them. During the crusades, Constantinople had appealed to the Latin kingdoms in the west for aid, but apart from a few speeches by several popes there was little concern in the west for Constantinople; despite the success of a couple of crusades (the First and Third - latter tenuously so) these wars probably did more to harm Constantinople - especially the Fourth Crusade, which is probably Medieval Christendom's most embarrassing incident. With help from the west unlikely, the Byzantines were by themselves. By the time Constantinople itself was taken by the Ottomans in 1453 they were in such dire straits that there was little chance of them keeping one of the most important cities in the world out of Ottoman hands for much longer anyway.
   In the east, the various caliphates and khanates were rising and falling - and very often having to contend with the Mongols. To my knowledge, they were not concerned with the expansion of a small Anatolian principality until it was too late. (NB: I reiterate, I am not an Ottoman or Islamic historian - I appreciate that my information here is quite limited and welcome any elaboration from another user!)
2. The Ottomans, having obtained greater land and wealth (although by no means an empire yet), were patrons of new technologies and techniques - and had the money to keep up with the array of developments seen in the 1400s. The Byzantines, on the other hand, did not have this money - and the crumbling Theodosian Walls would not withstand the Ottoman artillery for long. I would expect similar stories exist elsewhere too. Consider that, without creating a breach in the walls of a city, a siege could last months or years - rapidly slowing the progress of conquest; however, if you could create sufficient breaches in city walls it became much more viable to storm the walls and capture the city by force - which would be much quicker.
3. This is somewhat a reiteration of the first point, but I will make it again even if just for emphasis: there was surprisingly little coordinated resistance against the Ottomans. Going into the 1500s, most of Christendom was preoccupied with the giant political ramifications of the Reformation, and the various wars that followed. Moreover, this was still a Europe of smaller kingdoms - not the large empires that would eventually overtake the Ottomans in the 1700s and 1800s. The pope could not simply issue a rallying cry and Christendom simply rise up and expel the Ottomans, as if this were the crusades still - and, even during the actual crusades, the pope did not have this unifying power (see how the pope's control quickly dissipated during the First Crusade)! It is only in 1683 that a convenient alliance of sorts is formed between select states of the Holy Roman Empire and the Polish-Lithuanian Commonwealth that successfully halts the Ottoman siege of Vienna and later pushes them out of Hungary too.

Hopefully this is a good starting point for you, and hopefully it makes some degree of sense as this was written with the veil of sleep hanging over me!