Minorities?
Sunni and Shia are about 40 and 60 percent respectively so speaking of minorities doesn't really apply.
Kurds are Sunni mostly, but belong to a different branch, the Shafi'i, but also Sufis who are less strict. Shia are predominantly in the south including Baghdad; Sunnis in the central and northern areas.
Although the military under Saddam was Sunni-controlled, almost three-fourths of the lower ranks were Shia.

The Ottomans had three districts in what became Iraq, but Sunnis had the most power. The British continued Sunni dominance. The flip to Shia majority rule in the government erected by Bush only continued the frictions in a different direction; central control much weakened.

With or without the U. S., the strife will continue until massive relocation and massive refugee movements reduce the contact between the contending groups.

Talk of a 'democratic' government is nonsense when religion dominates.