When heat is supplied to water, for example, the temperature of water increases with the rise in heat energy but when the temperature reaches $0^\circ$centigrade or $100^\circ$centigrade, the heat energy is used to counter the intermolecular force and the temperature remains constant until all the ice has converted to water or all the water has converted to vapour. My question is that why is the heat energy not used to counter the intermolecular force when the temperature is not at melting or boiling point? Why does the vibration or the temperature keep on increasing even if the intermolecular force is not countered?
The key is the vapor pressure, which increases with increasing temperature and reaches 1 atm (760 mm Hg or torr) at 100°C:
This broadly corresponds to the broken molecular bonds that you refer to. Below 100°C, vapor bubbles, as the physical manifestation of the thermodynamic phase transition, simply can't resist the mechanical force of atmospheric pressure and therefore have a volume of zero. Above 100°C, vapor bubbles grow spontaneously, of course, as the effective internal pressure exceeds atmospheric pressure.
If the heat energy countered the intermolecular forces, let say at $80^\circ$ celsius, then $80^\circ$ celsius would be considered as boiling point. At $100^\circ$ Celsius the heat energy is very strong so that it can break the intermolecular forces between the molecules.