I'll try to answer your question, sorry if it doesn't satisfy you. I would primarily address the outbreak that happened in Ottoman territory, so no analysis on British-occupied Syria and Iraq (which we have little information about too)

The Spanish influenza affects Istanbul on July 1918. At first, the media shrug off the outbreak as unharmful. The Ati newspaper claimed the disease as benign and all its harm was that it distracted patients from their work for a few days. Similarly, Ikdam claimed that despite its spread, the disease was 'not pernicious at all' and that, 'like all infectious diseases', the disease required no medication for treatment and the patient solely needed to have a rest for recovery. However, when news about large numbers of deaths in Germany and Switzerland reached the Empire in August, Ikdam confessed that they were wrong and give measures to be taken and the treatment. While the disease was not as severe as in Europe, it still took a large toll and cause grief among the empire's populace

Effects on the Army

The epidemic first broke out in the Ottoman army on 6 August at the Army Station hospital in Nusaybin. This disease infected patients, doctors, and other hospital personnel, who experienced a burning sensation in the nose and throat, followed by sneezing, coughing, and high fever. It was carried by railway workers to the area, and despite protective measures, since the disease is airborne and could be transferred easily the disease spread to the central troops in Mosul on 15 August, then to the Tigris army in Tikrit on 25 August, and to the troops in Kirkuk and to the troops nearby on 26 August. The disease later on then reached Beirut on October and the Arabian Peninsula on December. In Beirut, More than half of the 17 soldiers of the German Asia Corps died between December 1918 and January 1919 due to the flu. In the Arabian Peninsula, the Spanish flu also inflicted casualties on the Ottoman garrison in Medina. In December 1918 Spanish flu suddenly broke out. Out of 11,000 men, 850 died because of the flu. Nearly 450 men died in January 1919. Even the commander, Fakhri Pasha caught the flu but later recovered. Emin Bey, an Ottoman officer in 1919 gave the total daily loss from the flu as 150, and calculated that the entire army will be decimated in two months. After the surrender and evacuation of the troops, it was reported that another 100 succumbed, but it's unclear whether or not they died

However, despite the losses and the scope of the pandemic, there were no significant complications and the number of casualties was low as the death ratio from the disease remained at a level of about 3.5%, and in case of Medina garrison, 11.8%. This was explained by the fact that the disease occurred in August, the hottest month of the year and the decline in humidity helped the disease lose its strength

Effects on civilians

Compared to the army, the civilians suffered large deaths due to influenza especially in Istanbul, where it was rather well-documented. In Istanbul, the epidemic comes in three waves, July to August 1918, October to December 1918, and January to May 1919. The first wave was rather benign, with cases diminished following a severe spread, and most cases were without any complications and there were few cases of mortality, probably due to the hot summer. However, the second wave hit in October, corresponding with winter and caused severe complications and deaths. Bronchopneumonia, pneumonia and pulmonary congestion were the complications often observed in patients, and bronchopneumonia, which was extraordinarily hazardous, was mostly the cause of deaths. Usually on the third or fourth day of sickness, the health condition of the patient suddenly gets worse and the fever gets even higher, breathing becomes difficult and a sticky phlegm starts to come out when coughing. Patients have pain in the chest and they start wheezing. This situation leads to death of the patient after 12, or sometimes 24-36 hour. Dr Akil Muhtar notes that he did not observe any case of recovery from severe complications. On December, the number of the death increased and there were approximately 400 deaths per week that schools were closed down until a further notice. Certain Armenian and Greek schools were reported to continue education despite the closure decision. The press publications of the time advocated for the implementation of quarantine measures regardless of sex and nationality and called for the immediate closure of such schools. Overall, Dr Hüsameddin Şerif suggested a number of 33,615 deaths by flu due to recurring waves in 1918. By late December the outbreak somewhat died down, the number of deaths from the Spanish flu decreased in Istanbul and thus it was now possible to open the schools again. However, other regions still suffered from the Spanish flu. In the Lapseki region of Çanakkale, many people died due to the fact that the central prison house was too small and did not have adequate sanitary conditions. In Sorgun and Kahta, 1160 died due to Spanish flu. In January 1919, the epidemic started to surge again, and although the 1919 wave is reported to be milder than the second wave, it was reported that the number of deaths was also high in the 1919 wave. Hüsameddin Şerif suggested 20,000 died due to the flu in 1919-1920, but it was unknown how high the number was in the January wave. Unfortunately, we have little data about the flu outbreak and the deaths attributed to it in the Ottoman Empire due to the lack of diagnostic evidence and considerable discrepancies.

Sources:

The Ottoman Army, 1914-1918: Disease and Death on The Battlefield by Hikmet Özdemir

Last Tango of the “Sick Man of Europe” with the Spanish Lady by Berna Arda and Ahmet Acıduman

The Impact of the 1918 Spanish Influenza Pandemic on Greater Syria by Kjell Jostein Langfeldt Lind