Most visitors come to Topkapi for the treasury and leave remembering the views. The palace sits on the point where the Bosphorus, the Golden Horn and the Sea of Marmara meet, and the terraces at its eastern edge look straight down over all three.
Inside, the sequence of four courtyards is the thing to follow. Each gate narrows the space and the crowd, and by the time you reach the inner courts the noise of Sultanahmet has gone. The Harem, entered separately, is the quietest part of all — small rooms, tiled walls, low light.
Allow half a day and start early. From Romance Istanbul Hotel the walk takes about ten minutes, so it is easy to arrive before the tour groups and have the first courtyard almost to yourself.