Suspension bridges are designed to absorb these forces in two ways: dissipation and transference. When a bridge dissipates force it spreads it out over a wider area. Transference involves moving force from a weak area to a stronger area.
Chances are you've seen a suspension bridge and maybe even crossed one. They consist of at least two large towers anchored to the ground, a roadway, and a web of steel cables strung between the towers and from each tower to the anchorages at either end. What's amazing is that the roadway is literally suspended in the air by the cables. It just hangs there as you and thousands of others drive across.
As the roadway hangs from the towers and vehicles move across its surface, the cables transfer the compression forces directly to the two towers, which in turn transfer the brunt of the pressure down into the ground. The cables also absorb much of the tension forces as they stretch under the weight of the roadway and traveling vehicles. Finally, the anchorages absorb some of the force as well.