no body supported hazrat ali nor hazrat amir muavia. as it was a political conflict rather than relegious. both have their own opinion and acted according to their judgement.
It was a political conflict but it involved very huge religious figures, including thousands of companions of Prophet. Hence it was more than just another political conflict.
Besides, that "political" conflict showed the extremely low character of Muawiyah and very high moral character of Imam Ali.
In this conflict, Imam Ali always took moral decisions and showed why he was heir to Prophet. On the other hand, Muawiyah used every scheme he could use to defeat Muslim caliph. His struggle was a personal one and not for Islam, because at his death he made his son his successor, and thus attempted to bring monarchy among Muslims instead of khilafat.
But I understand why you guys have to support Muawiyah's rebellion against the true caliph. After all, you took your Islam from Muawiyah. So even though you can not justify his crimes against Islam's caliph, you can atleast play dumb and ignorant.