I think you're confusing weight with bulk. While muscle is heavy, so is fat. How large your muscles get depends on how much weight you lift, yes, but also on the number of reps you do. The more work you do, the more you damage your muscles, and as they repair themselves they get bigger. How much bigger to the eye? That depends on genetics if you're a natural weight-lifter -- some people get really big muscles easily and for others they just don't. The size of the visible muscle doesn't measure the amount you can lift -- that's dependent on actual strength and also on intangibles such as desire and motivation. But if you're talking about power-lifting, which is when you lift very heavy weight, then you're probably going to bulk up a lot and you're going to look bulky. But that's not fat. What you might be seeing is, larger people are often the ones you see lifting weights all the time at the gym, and at a gym, those people are often overweight. They're also quite strong. The benefit of weightlifting isn't different than doing any physical activity -- you burn calories, you work your body, etc. If you don't want large muscles, lift less weight. Now, there is another class of bodybuilders who lift weights because they want to look highly muscled -- it's their goal. They often take steroids and growth hormone and eat diets far too high in protein and nutrients that help them look like that. Natural lifters can't compete with chemical lifters in how they look but they might actually be stronger.
So, you are talking about more serious weight lifting, I take it. Do you also have a cardio workout you do?