Not really, no. Enterprise actually turned hard to starboard away from Reliant, while Reliant remained on course. Really, it was ENTERPRISE that turned on its axis, not Reliant; probably the ship had to yaw to starboard using thrusters before using its impulse engines to change directions.
I can't picture how that would allow Enterprise to get in front of Reliant, for her port side phasers to hit Reliant from the front. Unless Reliant turned as well.
What's so hard about this?
1) Both ships moving towards each other head on.
2) Kirk sees Reliant rushing towards him in the viewscreen.
3) Enterprise turns hard to starboard (showing reliant her port side). Reliant--which does not turn--fires phasers, striking Enterprise' port side.
4) Enterprise fires back with port side phasers, striking Reliant--who has still not turned--with a front-aspect shot.
Really, it's as if Enterprise "crossed the T" with reliant and fired a broadside while Reliant maintained course. If Reliant turned at all in this exchange, Enterprise would have been firing on her from a side aspect.