I have moved the boundary of Fatih Mah. to include 64. Sokak (top left of https://www.openstreetmap.org/relation/7329589)) but when I search it still shows as in Reşadiye Mahallesi and with the wrong postcode - the postcode should be 55050.
Any idea how I can correct it to give FedEx a better chance of finding it?
The fact that one of the points of this road is still in the other district might cause some problems.
You could contact the Nominatim developers and ask them if there is anything you can do.
Of course, other software and apps (OsmAnd, Maps.me, Magic Earth, etc.) might already do the right job.
Did you check those ? They all have free versions, but map updates are only done on a monthly basis, so your latest changes might not be included yet.