Gulhane Park is the quietest green space in the old city, and the reason is historical: it was the private garden of Topkapi Palace until 1912, screened from the streets by the same wall that still runs along its eastern side.
It is a good place to slow down. The main avenue is shaded by plane trees for its whole length, benches face the water at the lower end, and the tea garden near the top gate is where locals sit rather than tourists. In April the tulip planting is worth a detour on its own.
There is no entrance fee and the gates open early. From Romance Istanbul Hotel the walk takes only a few minutes, which makes the park an easy way to start a morning before the museums open, or to end an afternoon once the crowds have thinned.