If you thought the crowds for Basilica Cistern and the Blue Mosque were "horrifying", just wait for Topkapı where once you finally make it to the innermost courtyard you face long lines to enter every over-crowded interior space :(
I don't know if any one day of the week is much worse than any other. In general though the more and larger the cruise ships moored near the entrance to the Golden Horn (neat Galata Bridge) the worse the crowds will be. The only time saving measure I know of is advance purchase of tickets but you'll only save time by not waiting to buy the tickets at the site itself.
Two years ago in the late fall I was trying to take a friend on their first visit to Turkey to Topkapı. We went three different general times (near opening time, around lunch and near closing time) on two different days to find the crowds impossible to bear. We went other places instead. Five or six years ago on a rainy day in November was the only time I found the crowds at Topkapı to be reasonable--it was also when I came to understand the Turkish word "üzgün". "Üzgün" is a curious word that is sort of like "rueful", "world weary", "ennui", "dejected" and "melancholy" combined.
If the weather is nice on Saturday or Sunday I suggest avoiding anyplace whose main attraction is based on being outdoors like a park, island, forest etc. as they will be crammed full of people who live in Istanbul trying to get "away".