There's a very straightforward answer to this question: Water has a much, much higher capability to transfer heat than oil.
Even if you were able to overcome the safety and design issues brought up in other answers, you'd still have to overcome the fact that water has a huge performance advantage at transferring heat compared to other common fluids (including oil). Water is essentially the best possible fluid in terms of thermal conductivity; the only fluids that beat it are metals in liquid state (which isn't a practical choice for cooling a car engine made from metal itself). At 75 C, water's thermal conductivity is around .65 W/mK. Typical heat transfer oils are usually around .14 W/mK.
Of course, coolant in a typical automobile isn't water, it's usually water mixed with alcohol. Typical mixes cause the conductivity to drop, because water also beats alcohol at heat transfer, but even pure ethylene glycol is .27 W/mK. A typical 50/50 mix puts you in the neighborhood of .45 W/mK, which means that typical water-based coolant has a thermal conductivity three times better than purpose-designed heat transfer oil.
Let that sink in. If you wanted to cool your car's engine with heat transfer oil instead of typical water-based coolant, you're operating at a three to one disadvantage in terms of thermal conductivity. This basically means that your coolant system needs to be three times bigger in order to have the same capacity. Radiator size is already somewhat of a limiting factor in terms of how the front end of a vehicle is designed. I don't think anyone wants to drive a car where the radiator needs to be three times bigger than a typical car radiator is today.
Oil is really only used for thermal transfer when water is impractical - due to corrosion, conductivity, evaporation, pressure or temperature requirements above what water can handle, or other limiting factors. As pointed out in other questions on mechanics.SE related to air vs water cooling, the transition to water cooled engines was basically because of this fact. An "air cooled" engine - which typically relies not only on air cooling, but also oil cooling - just can't compete. For a given application, if relying on oil or air cooling, you have to really limit your thermal load. In other words, for a given package size, it's easier to get more performance and efficiency with water cooling than with oil and/or air cooling.