"true love", compassion and suffering
Don't people who are more compassionate suffer more because they
  experience the suffering of other people?


The above appears to be the Western/Christian meaning of compassion, namely, "to suffer with". 

In Buddhism, the term "karuṇā" (often translated as "compassion") is the wish to ending suffering. It does not mean "to suffer with". 


  karuṇā - “ahita-dukkh-âpanaya-kāmatā,” the desire of removing bane and sorrow (from one’s fellowmen)