Hi Irina,
I have never really heard "car rent", but I've heard "car rental" (the activity of renting cars), "rental car" (a car which has been rented), or "to rent a car" (the action of renting a car). "Hire" can be used instead of "rent". A "lease" is another form of renting car, usually longer term (3 years), and often through a business/company (as it is tax-deductible, whereas a car purchase isn't).