there's precedent when it comes to a culture war between the Faith Militant and the royal family.

http://grantland.com/hollywood-prospectus/ask-the-maester-the-tale-of-brave-ser-barristan-the-faith-militant-and-the-fighting-harpies/

This column goes deeper into it, but the gist is that a couple centuries prior the Targaryens in charge wanted to do Targaryen things; ie, marry themselves. The Faith Militant revolted, turning one of their temples in King's Landing into a fortress. An assassination attempt on the king and his family almost succeeded, another Targaryen took his place (Maegor the Cruel), who set about committing virtual religion-icide, by mashing on the Faith Militant in every ugly way possible. That war resulted in thousands of deaths on both sides, eventually leading to a peace treaty between Jahaerys Targaryen and the Faith Militant to put down their arms if the king pledged to protect their people.

That peace lasted about 150 years... until Cersei came along. So yes, it may seem like the Gold Cloaks and the Tyrells have a stronger army, but this potential war isn't being fought in an open field, it's happening in the streets of King's Landing... where the common folk have the advantage.