What would be a way to get the modern population density of the fertile crescent under the Ottomans in the late 9th/early 20th century?
Iraq needs to be better connected to the rest of the Empire. Keeping Egypt and building the Suez canal themselves would do it, as would an earlier Suez canal. Or an earlier extension of the railroads to Baghdad (though railroads being built in Iraq requires the security situation to be solved earlier, and why build a railroad to a poor backwater province anyway - needs to be a military and economic case for short term benefits, else the Empire won't last long enough to enjoy the payoff).
Solving the security problems in the country also helps (18th and early 19th Century, the Ottoman territories were overrun by warlords and bandit kings - many independence movements would actually start as revolts against these parasites as well, so the central government regaining control of internal security earlier also means much less "push factor" to regions slipping from central control). In OTL, Iraq started to recover once the central government had broken the tribes and the warlords, so if that recovery starts earlier, population should get higher before the Empire runs out of time.
Solving the security problem means you need an earlier crushing of the Janissaries, since the mess that the Janissaries had become (basically, buying a Janissary pay stub was a way for Muslims to become tax immune and collecting pay from fictional Janissaries was a way for someone who owned a few dozen or hundred pay stubs to extract money from the government) was deeply interwoven with the system these provincial warlords existed in.
EDIT: Without a PoD in at least the 16th Century (such as no Safavids), modern population densities for Iraq aren't happening by the time limit you give. The above changes could maybe make Iraq competitive with Syria in terms of per capita productivity and population/acre of farmland.
Because they lacked funding. Traditional Ottoman medicine system and folk medicine also had strong support among the local population.
They lacked funding to support even local best-practice. The state of the charitable foundations that supported public medicine in the Ottoman empire had decayed badly in the 18th Century. Ottoman charities worked by having a pool of assets linked with a charitable cause - so a hospital might have several villages that paid rents to the hospital as their feudal protector, and those rents would pay for the operation of the hospital. However, the managers of these charities were always trying to pass on management to the next generation of their family especially since charitable foundations weren't taxed, so if a family could gain control of a charity's assets they could transform it into a false charity - cut the budget of the hospital, amass tax-free wealth that wasn't technically inherited by the next generation (Ottoman inheritance law was not friendly to generations passing too much wealth on to the next) and voilà! You have an immortal feudal parasite under the control of a powerful family and a severe lack of public infrastructure and a severe lack of a tax base. The struggle between parasites and those trying to keep Muslim charities honest is an old one, and the Ottoman empire went through several ups and downs in this cycle, but the 18th Century had seen an especially long and severe absence of the state restoring sanity and justice to the system.
In short: in the economic sphere they preferred free markets until it was too late.
"An Economic and Social History of the Ottoman Empire" edited by Halil İnalcık differs.
It isn't clear that the Ottoman Empire suffered from having a free market. In the first place, it meant that Western traders didn't have to fight wars to get access to the Ottoman market - which if you remember didn't end well for China. To the contrary, Westerners were given a reason to desire the integrity of the Ottoman economic unit to ensure their trade and investments did well. Nor was this a story of only Western success building on Western success - while foreign trade was quickly out-competed by the French starting in the 17th Century (who in turn were out-competed by the British, who in turn were in the process of being replaced by a number of second-wave industrial powers when the Empire fell) - this doesn't really harm the Empire, or its merchant class, who take advantage of the expanding internal trade driven by the rising external trade. That foreign trade is being sent out in foreign owned ships doesn't lead to a rolling takeover of the rest of the Ottoman economy. Further, while the free market does lead to some industries being pretty well wiped out (yarn spinning on the coasts for example), import of cheap yarn from the UK and synthetic dyes from Germany lead to a great expansion of weaving, dying of plain cloth imported from England and of rug making for export (not an industry the Ottomans had been much involved in before), while artisans on the coasts are on the whole slowly declining in the face of industrial imports, the growing of cash crops for exports is a major driver of peasant prosperity.
Further, had the Ottomans opted for protectionism and invested heavily in building their own factories, they'd have been building them in Trace, Bulgaria and Northern Greece, not in Anatolia and Syria. Not very useful unless they can solve the problem of losing their core territory.
And if they can hold on to their territorial core over the 19th Century, they keep all the places where their first railroads and factories had been built and can build on those early efforts, rather than having to start from scratch after every time they lost part of the Balkans.
Also, the Balkans are also where most of the Empire's decent coal is. Anatolia has some coal, but it's power station coal, not blast furnace coal.
As such, I think secure borders with a free market and we could see an Ottoman industrial revolution in time, with a closed market and insecure border I think they actually do even worse than OTL. With secure borders and a closed market
might be better, but I am doubtful. Either way, first thing is for the Ottomans to be militarily strong enough or lucky enough that they aren't slowly being eaten alive for their last century is key to them becoming more developed than OTL.
Add to the territorial security issue that the Ottomans just being a little too slow off the mark to start serious reform (smashing the Janissaries even a decade earlier would have been a great help I think, to say nothing of even earlier), being Muslim and competing against a continent otherwise made up of Christian states, recovering from an extremely bad 17th Century and I think you have a better explanation of Ottoman economic performance.
fasquardon