Tybalt is fired up about Romeo being at the Montague's party. Tybalt is completely ready to fight Romeo and can't wait to completely let go of control and have it out on Romeo. When Tybalt approaches anyone in the play, within a page, there is a fight or the desire from Tybalt to get into one.
This was the case in Act III, scene i when Tybalt approached Mercutio and Benvolio looking for Romeo. When Romeo arrives later, Romeo wants to explain to Tybalt why he can't fight him, but Tybalt is ready to explode. This is the moment in which conflict intensifies. The secret has to be kept for now. Romeo can't explain to Tybalt that they are family now. If Tybalt could have heard that things might have been different, but because he couldn't things got out of hand... twice.
In Shakespeare's Romeo and Juliet when Romeo arrives during the confrontation between Mercutio and Tybalt, his attempts to arbitrate and ameliorate the animosity between Tybalt and him causes more problems. For, Tybalt is enraged at Romeo for having the effrontery to say that he loves him:
Romeo, the hate I bear thee can afford/No better term than this--thou art a villain.
Tybalt, the reason that I have to love thee/Doth much excuse the appertaining rage./To such a greeting. Villain am I none,/Therefore frewell. I see thou know'st me not.
Boy, this shall not excuse the injuries/That thou hast done me, therefore, turn and draw. (III,i,49-55)
Then, this choleric outcry, added to what he perceives as weakness in Romeo, causes Mercutio to become incensed:
Oh, clam, dishonorable, vile submission!/Alla stoccata carries it away. [Draws a sword] (III,i,61-62)
Thus, the well-meaning intervention of Romeo certainly goes awry and causes more conflict. Even later, when Juliet learns that her beloved cousin Tybalt has been killed by the hand of Romeo, Juliet is conflicted in her grief and torn between her affections for both men.
I am really not convinced that the marriage does intensify the conflict. I guess you could argue that it does, but only sort of indirectly. Here's how I would argue it:
After the marriage, Romeo is full of love and feels like he is part of Juliet's family. Because of that, he does not want to fight Tybalt. Tybalt still hates him, though. Because Tybalt wants to fight and Romeo does not, Tybalt manages to kill Mercutio (because Romeo is trying to hold Mercutio back). This leads to Romeo killing Tybalt.
So you could say that marriage makes romeo not want to fight which makes Mercutio die. This intensifies the conflict, but not for long.