There's ample evidence for Romeo's immaturity in Romeo and Juliet; in fact, it's arguably the flaw that leads to his ultimate downfall.
Romeo immediately "shows his cards" with his fickle behavior. In Act One, Scene One of the play, Romeo complains to his friend Benvolio that he is desperately in love with Rosaline, a woman who has decided to take up a vow of chastity. However, by the end of this same Act, Romeo has forgotten all thoughts of Rosaline in favor of his new obsession: the young, lovely Juliet. After spotting Juliet at the Capulet's ball, Romeo proclaims:
Did my heart love till now? Forswear it, sight! / For I ne'er saw true beauty till this night.
This moment is multilayered. First, the fact that Romeo is capable of shifting such significant emotions so quickly suggests that those emotions are lacking in true depth. Second, Romeo's claim that he had never loved until the moment he saw Juliet implies that he is not in possession of self-awareness. Finally, Romeo chooses two women who are quite plainly poor choices for him due to their unavailability. Rosaline is celibate, while Juliet is a member of Romeo's rivaling family.
Romeo can also be deemed immature because of his impulsive behavior throughout the rest of the play. He kills his love's cousin, Tybalt, in order to avenge the death of Mercutio, knowing that this action could create an enormous rift in his relationship and put his status in Verona at risk. Indeed, it does; although Juliet is able to forgive him, Romeo is banished from the city. When he returns to Verona to discover that Juliet is "dead," he is quick to take drastic action; he consumes poison, dying in her crypt before she can re-awaken.
Ultimately, Romeo's immaturity is displayed through his persistently poor choices, his rapidly shifting emotional landscape, and his inability to contain his violent behaviors.