Now US breeders have produced a blue colored Green Iguana. Green iguanas' are named Iguana iguana. They all, well 90% of them, have the trait of being highly resistive to human handling. Before being tamed, they nearly all have the behavior you describe. If you wait more than a month or two after they're hatched, they fight you exactly like older iguanas, except they are too small to break your skin. This is the time to train them.
If yours is older than that, depending on it's strength, it can hurt you, and un-tame adults can slash you open and even remove a finger if it gets ahold of it just right.
Descriptions (sp) of the taming method for these reptiles are kind of long, but are simple to perform, so if you want a description, just post again and ask for it.
For now, you've given me one or two clues on how can do better. First, you have the side approach right, but never do it slowly. It just builds suspense, making the attack more frantic. As most of us here advise, reach in with a purpose, swiftly and firmly grabbing it by it's front shoulders and quickly pulling it out and grabbing it's hips. Your ig is not delicate, so use enough force of overcome it. Press it against your stomach hard enough to stifle it's ability to bite and lunge out of your hands. He will struggle for thirty seconds and then hold still. Take it to a chair of sofa and spend fifteen minutes holding it carefully, being very careful to not let it get away. Do that at least twice per day at first.
Broken tails are normally caused by the iguana flailing it's tail and contacting a hard surface such as part of the cage when it's on it's way out. You may have to use your other hand to control it's tail as you take it out.