General Synonyms:
* Appease
* Reconcile
* Pacify
* Sooth
* Propitiate
* Placate
* Mitigate
* Assuage
* Harmonize
* Mediate
* Compromise
More Formal Synonyms:
* Ameliorate
* Conciliate (can be used as a synonym for itself)
* Expiate
* Mollify
Less Formal Synonyms:
* Smooth over
* Make up
* Make peace
* Bridge the gap
The best synonym for "conciliate" will depend on the context and the specific meaning you want to convey.