I went to Belgrad Forest for a run... There is a nice 6km track around a lake with some climbs and some descents that will break the monothony. There is enough people to make you feel safe but not that many that you can't run quietly.
If you don't have a car, getting there is not complicated: you take the subway to Haciosman and take a cab (~20TL) to the park's entrance; getting out however might be more troublesome - I was lucky because a cab had passed by after dropping someone.
There is good water to drink and some fitness equipment around the track.
There are places for pic-nics and also dirt bike tracks.
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