Thurio is a nobleman at the Duke's court in Milan. He's the Duke's favorite to marry Silvia, which makes him a rival to Valentine. Thurio is kind of arrogant and annoying, so he's not much competition for Valentine, even though he tries really hard to win Silvia. Like some of the other characters, Thurio is a tad bit fickle. He loses interest in Silvia as soon as she runs away to the forest to find Valentine:
Why, this it is to be a peevish girl,
That flies her fortune when it follows her.
I'll after, more to be revenged on Eglamour
Than for the love of reckless Silvia. (5.2.13)
We can understand why he doesn't want to be with a girl who is in love with another man, but, then again, the fact Silvia is in love with Valentine never seemed to stop him from pursuing her before. It seems like Thurio loses interest in Silvia because she's a disobedient daughter, which, in Thurio's mind, means that she'll also be a "disobedient" wife.
Thurio also seems a bit petty, especially when he says he's going to tag along with the Duke to the forest just so he can see Eglamour get in trouble for helping Silvia run away.