Istanbul is a "World City" in every sense of the phrase. The closest city with a larger population is Karachi, and while population isn't the ONLY measure of "World City" status, it is a significant factor. When you couple that with it's cultural heritage, it's location as the gateway between Europe and Asia, and it's economic importance - there can't be much doubt about it.

Now, it is true that it is no longer the most important city in Europe, but it one was during the late Roman (Byzantine) era when the capitol was moved from Rome to Constantinople. It was the capital of the only empire in Europe, so it's prominence was heightened because no other city in the region even came close.

Over time, as western Europe transitioned out of the dark ages and Islam took hold in the middle east and Africa, political power decentralized away from Constantinople.

Putting aside for the moment of what you mean by "world city", one thing to look into in terms of 20th century changes is the decision to establish the capital of the Turkish Republic to Ankara. Ataturk's decision to do so was driven primarily by the need to dissociate the new Republic from all the imperial and dynastic trappings, history, and culture embedded into the old imperial capital.^1 Would Istanbul have remained your "world city" had it been kept as Turkey's capital? That's a counterfactual, as undesirable as it is, whose answer may be the best way of answer your question in the 20th century.

^1. summarized from lecture notes, which may or may not have been from the following articles:

Sinem Türkoğlu Önge, “Spatial Representation of Power: Making the Urban Space of Ankara in the Early Republican Period,” in Power and Culture: Identity, Ideology and Representation, edited by Jonathan Osmand and Ausma Cimdina (Pisa: Plus-Pisa University Press, 2007), pp. 71-94

Zeynep Kezer, “Contesting Urban Space in Early Republican Ankara,” Journal of Architectural Education, vol. 52, no.1 (1998), pp. 11-19

Biray Kolluoğlu-Kırlı, "Forgetting the Smyrna Fire," History Workshop Journal, no. 60 (2005), pp.25-44

Well, it still is and nearly always has been since Constantine. Up to the point of the Fourth Crusade it was the largest city in Europe; that disaster in 1204 reduced it to a small town over the next two centuries. Then, under the Ottomans it gradually regained its status where by the 16th century it was again one of the biggest cities in Europe and a hub of international trade. The Ottomans and their capitol played major roles in European politics to this day.

I think it's just not mentioned as much as it used to in the West, but it's the same city it's always been.