You can lose weight quickly, but not safely. And the weight that was lost quickly, returns just as fast.
The ketogenic (no-carb) diet is one of the fastest I've seen - personally I lost 10kg in one month, and two weeks afterwards, 6kg back on. You don't have to use the high protein, no carb, additive laden 'supplements' that cost a fortune. But it puts an incredibly strain on kidneys, and must not be used for longer than a couple of months. It's very hard to stick to, and I did not feel well on it, despite the reassurances of the naturopath who asked me to do it as part of a treatment plan for chronic pain. Incidentally, I lost my ability to think clearly - important for working. I would recommend strongly *against* it. Carbohydrates are necessary - everything in moderation.
The best way, is to eat healthily, move regularly, sleep lots and drink heaps of water. Daily calories should be in deficit - tracking everything you eat, drink and do helps to stay on track.