When to come — and when to absolutely avoid
By season
Quietest: Late November to early March, excluding the New Year week and the Turkish school half-term in late January. Istanbul winters are mild (8–12°C), the cruise crowds are gone, and the queues melt away.
Busiest: Late April through August, with the absolute peak in July when Sultanahmet is overrun by tour groups, school holidays and cruise-ship arrivals at Galataport. If you can possibly avoid these dates, do.
By day of the week
Tuesday and Wednesday are the quietest. Fridays are tricky because the ground floor closes from approximately noon to 2:30 PM for Jumu'ah prayer — but the upper gallery stays open throughout, so it's actually a strategic time to visit if you only care about the museum.
By time of day
The first slot at 09:00 and the last slots after 16:00 (winter) or 17:00 (summer) are noticeably quieter than the 11:00–14:00 peak when most cruise-ship and bus groups arrive. Late-afternoon light through the high windows is particularly beautiful on the gold mosaics.