Sultanahmet is the one walk in Istanbul that competes with Venice for crowds. The trick is timing. Arrive at Gülhane at opening — 7 or 8am depending on the season — and you have the sites almost to yourself for the first hour.
This route takes them in the order that works: outer garden first (Gülhane, which is free and open at dawn), then Topkapı's outer courtyard (also free), then Hagia Sophia right as it opens. By the time the coach groups arrive at Hagia Sophia you'll be in the Hippodrome, coffee in hand. The Blue Mosque closes for midday prayer — work it into the rhythm. End in the cool of the Cistern; two hours of sunlight after that and you're ready for a long lunch in Balat or a ferry to Kadıköy.