Question

Dose bar magnet exert a torque on itself due to its own field

Answer 1

Yes.

But in such a way that the sum of torque is zero. This is because while macroscopically the magnetic field is this

but individually, they are a sum of so many particles of that similar to given below

The electron here, would interact with another electron nearby. The electron interaction also makes the coupled atoms interact. So in a way magnetic fields to affect the bar magnet, but on a whole they can said to be in inertia of whatever they are in.