Then the problem becomes defining the borders of the voting units. Do you use the old Ottoman vilayets? The borders of the various League of Nations Mandate territories? Or do you appoint a commission to draw up relatively-homogeneous voting units? (Cue massive quarrels over composition of the commission.) Larger units will not be homogeneous, smaller ones will create massive balkanisation. In short, this could actually create as many problems as it solves, hard as that is to believe. It is true that, if you were going to be rational about it (i.e. try to make the borders run along existing ethnic and sectarian cleavages), there would scarcely be a border between the Mediterranean and the Hindu Kush that would be where it OTL is; but it is a lot easier said than done.
Let's stipulate, though, that you use the vilayets. Then the question becomes, what is the question i.e. what are you actually asking? You can ask the population of the Mosul vilayet, for instance, 'do you want to be independent, or united with Baghdad, or with some other combination of vilayets?' - there are too many possibilities to answer without getting every voter to write an essay. There is no way of framing the question that it would be universally seen as fair. Cue immediate conflicts over grievances related to exactly which question had been asked.
The British and French undoubtedly could have done better (i.e. shown less selfish motives), but there would certainly have been a lot of trouble in any event. So much, in fact, that it might not be possible to impose the referendum results without massive violence - which possibly defeats the purpose.