We know that starships can change directions, turn, bank and dogfight at warp speeds. This is visually demonstrated on the Enterprise's view screen in "The Wounded" TNG 4x12, when the Nebula Class starship Phoenix (captained by Starfleet Captain Benjamin Maxwell) and 'flying' at warp in close formation with the escort ship Enterprise, suddenly changes course and banks hard right to pursue a Cardassian ship.
The question is; How tightly can they turn? As example; can they orbit a planet at warp speed?
Picard, Data, Yar and Troi enter the battle bridge, with Chief Miles O'Brien manning the conn. First, Picard orders that Yar fire photon torpedoes towards the object. Yar complies and the torpedoes are away. Shortly after, Picard orders that the countdown to saucer separation begin. Data counts down, and the ship separates while at warp. The stardrive section turns around and heads towards a confrontation with Q.