I think that to have perfect 90° turn without encoders is next to impossible.
I'll explain why: Same make motors at same voltage will have slightly different speed and torque making robot to drift left/right a bit, that is, You will never make exact 90° turn unless uC knows how much each wheel have spun.
Regarding robot positioning just before turning:
| X X X X X |
1 2 3 4 5
While line has no sharp corners robot should be guided by sensors 2 and 4 with sensor 3 always detecting white line.
When robot approaches 90° turn on the left, sensors 1, 2, 3 and possibly 4 will detect white line. Then using encoders You measure width of the left turn line by moving robot forward until sensor 1 does not detect white line anymore. Assuming that distance from bot's wheelbase to IR sensor is more that width of white line, You calculate further distance Your bot needs to move by dividing line width by 2 and subtracting result from wheelbase-to-sensor distance; move Your bot forward by that distance and turn both wheels to opposite directions exactly by quarter of rotation.
To get more precise rotation without encoders You should probably use line sensor with more IR Emitter/Detector pairs in it, this will give You better resolution of how far from line centre Your robot is. However, I still see no way of having perfect 90° turn without encoders.