Why did Tywin order the sack of King’s Landing? Sacking the Red Keep itself makes sense, but why massacre & loot the common populace? The commoners certainly didn’t love “King Scab” enough to rise in his defence, and they had never done any harm to nor slighted House Lannister. Hell, they probably liked Tywin thanks to his trade promotion, tourney hosting ,road building. crackdown on food adulteration. Plus the Westermen were fresh, so no question of their “blood being up”.

Historically speaking, it’s incredibly difficult to restrain an army from sacking a city that falls during a siege. For example, Holy Roman Emperor Charles V, as devout a Catholic as one could find in the Reformation and the most powerful man in Europe, couldn’t stop his largely Catholic army from sacking Rome – massacring the Swiss Guard almost to a man on the steps of St. Peter’s, looting almost every church and shrine in the whole of the city, and killing between 6-12,000 civilians. And that wasn’t a case where his men had their blood up – Charles V hadn’t paid his men in a while, so they just took their wages in plunder. 

As for Tywin, he didn’t care. I doubt he gave an order – his orders were to eliminate the royal family, which was the main political priority – because as far as he’s concerned, the smallfolk do not matter. His men sacking the city allows him to avoid paying some wages, so why not?