I may be wrong, but it doesn't look like they controlled any culturally significant cities either (other than the Syrian cities that belong to the Ayyubids by the time this map is set)
That is just wrong. At that time in the middle ages the arabian world was much more prosperous, technologically advanced and therefore more densely populated than the western world. For instance the Kingdom of Sicily after the Norman conquest was more powerful that the Kingdom of Aengeland. Before the Conquest areas of the Kingdom were mainly Muslim or Byzantian, which led to its population growth.
Same goes for Al-Andalus in Spain. Under Muslim rule it flourished. Its economical decline came with the reconquista. Maghreb, Mauretania, Ifriqya and Egypt, all no exception. From Spain to India you had many major economical centers that dwarfed cities in western europe, all under Muslim rule.
In Spain:
Qurtuba (Cordoba) not as many people as under the Umayyads, reaching half a million, but still very important.
Qa'lat Garnata (Granada)
In the Maghreb:
Fes, with 150.000 to 200.000 inhabitants during the crusades.
Marrakech with 150.000
In Ifriqya:
Tunis (target of the seventh crusade)
In Egypt:
Cairo, 150.000 to 300.000 in that period. Very Important.
In Italy:
Palermo 150.000
for comparison:
Rome 20.000
Firenze Milan Genoa Venice around 50.000 getting more prosperous some hundred/ two hundred years later
In Western Europe:
Paris maybe less than 100.000, 150.000 hundred years after your time setting.
In the Seljuk Empire (persia):
Baghdad 250.000
Tabriz 100.000
That only were the major political and economical centres. Countless other Muslim cities were still bigger than most western cities.
Constantinople had around 250.000 at that time
And Jerusalem and Rome were only important as religous centres and symbols, not so much economically.
So if you think about important cities, with a few exceptions you think Muslim.
You have to understand how important irrigation techniques were at that time. There were times during the middle ages in which Japan managed to have a bigger population than whole Western Europe, thanks to advances in irrigation.
Edited 11/2/2014 12:58:25