Struts

Checking for oil leaks is the first step.
A bouncing test would be next followed by removing the struts and testing its damping force manually. A machine test for damping force would be the ideal way but there are not many places with such equipment as it is expensive and usually only manufacturers have such equipment.

Some poor quality tires would start cupping due to bad wheel alignment, balancing etc and when they start, they would go from bad to worse. Replacing with a good quality tire might resolve the problem.