It is possible to lose weight through exercise alone. I have lost maybe 1 ½ stone of fat in the last couple of years (I weigh one stone less than I did to start with but have gained a significant amount of muscle, so I reckon I have converted at least half a stone of fat into muscle).
However:
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I eat healthily anyway. Very little processed food and lots of fresh vegetables. The only thing I changed was to drop my portion size, as this was what had caused me to put on the excess weight.
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I was not hugely over weight to start with. I have gone from a size 12 back down to a trim size 10.
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The weight has come off slowly over about 2 years. I dropped half a stone in about 6 months this year when I intensified my training.
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I do a lot of exercise. I have been training for triathlon this year, so cycling for 7-9 hours per week (including commuting to work), running for 1 to 1 ½ hrs, and swimming for 1 to 1 ½ hrs.
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Because I have been burning around 6,000 - 8,000 extra calories a week, I do take on more than 2,000 per day, either by simply eating more, or by using energy supplements. It is not possible to train hard for 9+ hours per week without providing your body with extra fuel. This includes using energy supplements that are high in readily accessible sugars and carbohydrates.
So what I think this shows for you is that yes, it is possible to lose weight whilst eating a diet high in sugar and processed carbohydrate (whether in the form of cake and convenience food, or in the form of energy supplements) but you have to do a serious amount of exercise to counter it. The simplest and easiest way to lose weight is to look at your diet first: to eat healthy, non-processed, low GI foods in reasonable proportions. Doing exercise will get your metabolism working faster, and will motivate you to eat healthier, but will not on its own counter eating badly.
To go into the science of it a little more (and I have to say my knowledge is fairly rudimentary so I am happy to be corrected by anyone more knowledgeable), is that your body stores reserves of energy in the form of glycogen in your liver. This provides enough readily-accessible energy for around about an hour's worth of vigorous activity. If you do just an hour's exercise and then stop, you will need to replenish those stores of energy. It is really quite hard work for the body to break down fat to replenish those stores, and it finds it much easier to replenish them from the food you eat. If the next meal that you eat contains plenty of sugars and readily accessible processed carbohydrates, then the body finds it easy to replenish its energy stores. It has to work much harder (and therefore actually use energy/calories) to break down complex carbohydrates and proteins, and harder still to break down body fat.
If you are only exercising 2-3 times a week, burning maybe 300-400 calories a time, and you are eating a sugary diet high in processed carbohydrate, then your body will replenish its energy stores almost exclusively from your diet and won't start breaking down your body fat at all.