It is important to note that the conflict actually escalates in Act I scene 1 as law enforcers arrive. As an Officer and four Citizens with clubs enter to try and suppress the violence, then Lord Capulet and Lord Montague then enter, and try to get involved in the violence themselves, escalating a brawl between the houses to a duel between its two principal heads. It is only the arrival of the Prince with his train that actually manages to bring an end to this violence. Note what he says that manages to stop the brawling:
Rebellious subjects, enemies to peace,
Profaners of this neighbour-stained steel--
Will they not hear? What, ho! You men, you beasts,
That quench the fire of your pernicious rage
With purple fountains issuing from your veins!
On pain of otrture, from those bloody hands
Throw your mistempered weapons to the ground
And hear the sentence of your moved prince.
If we look carefully at these first words from the Prince, it is inferred that even his arrival is not enough to immediately stop the fighting. His question "Will they not hear?" strongly suggests that they keep on fighting in his presence. He is thus forced to threaten them with torture if they do not stop immediately. This of course indicates the depth of the rancour between the two houses.