You should be aggressive and fast even with the light weights. It's not about pulling just enough for you to drop underneath it; it's about elevating the bar as high as possible and then using the bar to pull yourself down into the squat. This way you're keeping constant, controlled tension on the bar instead of bumping it out and letting it float.
If you're getting bruises on the pubic bone then you're either leaving too much space between you and the bar (not pushing the bar all the way into the hip like you were saying) or you're initiating the second pull to early. From what I see, you might just need to push the bar into the hip a little more.