In my textbook, it is given that internal resistance of a car battery increases with decreasing temperature. But car batteries contain lead, which is a metal, and hence the resistance should decrease. Where am I wrong?
If the battery only contained lead, you might be on to something. However, batteries are made with both metals AND something to transport the charges, called an "electrolyte". This is normally a liquid, and liquids have less motility as they get colder. This motility refers to the speed with which something can move through the liquid.
Chemomechanics is right. Inside a battery, the dominant contribution to the internal resistance is not the lead plates but the powerful acid solution the lead plates are immersed in, and the boundary layer between the acid and the lead surfaces. Those resistivity effects are driven by chemical kinetics, most of which are slowed by lowering the ambient temperature and speeded up by raising the temperature.