What causes a lot of bouncing in the front while slowly driving over a speed bump?

Typically, it's either shocks or struts, depending on which your car has. Shocks/struts absorb the resonance energy of springs. The PURPOSE of the shock/strut is to stop the wheel from bouncing when you hit a bump. Without the shocks, the wheel would continue to bounce, sort of like a Basketball. When the shocks work properly the wheel is "dampened", keeping the tire in contact with the road and giving better control. The bonus is, the car doesn't bounce, the way you're describing. The good news is, they're not terribly expensive and they're usually not impossible for the DIY'er. But if you prefer, it's not terribly expensive, as auto repairs go. If you have someone do the shocks, have them check other suspension components at the same time. No point in replacing just the shocks if the whole front end is worn out.