This is my uneducated take :

Conservatives have been taught to follow the leaders in their environment, without questions (see religion)

It gives their leaders a lot of power and thus, there is a lot of incentives to become a leader through whatever means possible. It gives leaders power and influence.

So comes along an issue and a non-conservative authority tells people they should do X. Conservatives leaders want more power and they hinge on that thing to gain more visibility and thus more power.

The other conservatives see that their leader is against, and so they take their leaders stance without question and hate that thing too.

Conservatives leader will say whatever they can without really really caring if that is true or not (remind you of someone?)

It is a lot easier to rile up people with negative emotions and tribalism, so the conservative leaders use hate and fear to further their strength and influence.