Putting bread in the fridge does help prevent mold growth on the bread but it's not really recommended to put your bread in the refrigerator as the bread tends to go stale much more quickly in the fridge than it would when out at room temperature.
I would get a breadbox and put the bread inside the breadbox and use it up in a few days or so before it has a chance to grow mold on it.
The bread may not be as tasty if you refrigerate it and will be stale more quickly than just putting the bread in a breadbox or keep it out on the counter or in the cupboard.