You answered your own question, never EVER put water in the radiator, it'll kill your engine slowly, if not by exponentially overheating it in the summer and killing it instantaneously. Put coolant in it, and your problem will be gone. It's a trick most mechanics know but don't share cause they know most of them will then be out of a job. Suzuki makes a habit of putting tap water in their cars from the factory, though thankfully toyota and honda don't do so. Secondly if you've had water from the start both thermostat valve and radiator cap will have rust on them, so its better to just replace them. Get guard coolant concentrate and mix it with distilled water in a 70/30 ratio(since our temperatures don't go below -5C, I suggest 70/30 mixture). Open radiator cap and then remove radiator drain plug and drain all the water. Reinsert the drain cap. Empty the coolant reservoir bottle and drain it too. Start pouring the mixture, while pouring the mixture, have someone start the car so that the water pump can suck the coolant in. How do you know you have enough coolant? Extra coolant will start dripping from the overflow hose in the radiator and you'll see it dropping to the ground. Now fill in the reservoir bottle with the coolant a little bit above the min mark. That's it you're done.
Or instead watch this video, this guy makes it look super easy:
