How is magnetic field line diagram of a bar magnet found for region inside it? How can we prove that magnetic field lines inside a bar magnet will look like as shown in the first image? By what experiment did we find this? You may say we found this by placing cardboard on the magnet and then sprinkling iron filings on it and then, you say, and the pattern shown in the 2nd image emerged. Yes, but the pattern that emerged is representative of the magnetic field due to the magnet on the plane of cardboard and not of the field inside the magnet. This experiment cannot be used for proving that field inside will be as shown in the first image.
 
 A: How can we prove that magnetic field lines inside a bar magnet will look like as shown in the first image? By what experiment did we find this?
I am not aware of any experiment. At most, you could build a sandwich of bar magnet - paper - iron filings - paper - bar magnet. After removing the two uppermost layers, one looks again at the filings. I don't expect any other pattern than on your picture.
The more interesting consideration is why does the magnetic field distribute as shown by the iron filings. For this purpose, it is sufficient to consider the atoms or, even more in detail, the unpaired electrons of the magnetic material.
Each electron is at the same time an electric charge and a magnetic dipole. In some materials with unpaired electrons in the atoms, the atoms organize themselves in such a way that their magnetic dipoles have in summary orientation over the whole material and are a permanent magnet.
Thus, it becomes clear why the magnetic field looks the way it does when observed from the iron filings. The common alignment of the atomic dipoles lead to a largely homogeneous magnetic field in the material. However, since a magnetic field also chooses the path of least resistance, the magnetic field in bar magnets clearly emerges laterally to form shorter loops.
