Can I put purified water in my car engine

If, by "purified," you mean distilled water, you can use it to mix with coolant for your radiator and you should only use distilled water in your battery. If the water has been purified by some other means, it would depend on what's in it.

You can put purified water into a car battery, but it is better to use distilled water (not the same thing). 'Purified' simply refers to the absence or removal of contaminants which are harmful to humans, not batteries, so purified water will usually still contain trace minerals, etc. which can possibly cause harm to your battery.