It's important to note that when the play begins, Juliet is only thirteen years old. Regardless of era, that's an age barely out of childhood.

Through her actions, it's easy to see why so many believe she is an older teen (or a young woman). She is willing to disobey the wishes of her father, who has arranged an advantageous husband for her. Juliet stands her ground with her father, telling him that she cannot marry Paris, until he threatens to disown her. His tone in act 3, scene 5 is threatening, but Juliet does not waver in her convictions that Romeo is her choice. She turns to her mother for support, but she doesn't find any help there, either. Lady Capulet writes off her daughter, telling her that she's done with her. In desperation, Juliet looks to her faithful nurse, who tells Juliet that Paris isn't such a bad match after all—and maybe even better than Romeo, whom Juliet is already married to (thanks, in part, to the nurse herself).

At this point, most thirteen-year-olds would fold, having no support and, in this society, few other options if she didn't follow the wishes of her parents.

Juliet's response is to fake her own death. Just before doing so, she considers the possibility that the friar has actually arranged for her to drink a poison that will kill her so that his own hand in the marriage to Romeo is hidden. She thinks about the reality of laying in a crypt, waiting to wake up and hoping Romeo will come.

And still—she drinks the potion.

Juliet proves to have a fierce courage and maturity that is far beyond her thirteen years of life experience.

Arguably, Juliet demonstrates rational and mature thinking, but this maturity is, for the most part, demonstrated before she gives her heart to Romeo. 

In Act I, Scene 3 of Romeo and Juliet, Lady Capulet asks her daughter if she can consider Count Paris as a husband; she also informs Juliet that Paris will be at their feast. Juliet replies that she will take a look at Paris and try to like him if what she sees is worth liking. But, she adds that she will not let herself fall for Paris any more than her mother's permission allows. In other words, Juliet has a sense of moderation at this point:

I’ll look to like if looking liking move.

But no more deep will I endart mine eye

Than your consent gives strength to make it fly (1.3.99-101)

Certainly, this sense of moderation demonstrates a degree of maturity.

Further, in the famous balcony scene of Act II, Scene 2, when the passionate and impetuous Romeo swears his love and desire for Juliet, it is she who urges caution:

Oh, swear not by the moon, th' inconstant moon,
That monthly changes in her circled orb,
Lest that thy love prove likewise variable. (2.2.116-118)

Then, when Romeo asks what it is that he should swear by, she tells him not to swear at all. Instead, she urges caution, saying that her agreement to a betrothal is

...too rash, too unadvised, too sudden,
Too like the lightning, which doth cease to be
Ere one can say "It lightens." (2.2.118-120)

Juliet maturely adds that their love is like a bud that will prove to be a beautiful flower if it is allowed to ripen, so they must be patient.

Despite her rather mature and cautionary behavior in the beginning of their relationship, Juliet is later influenced by Romeo's passion, and her love transforms into a "violent delight." That is, the love of Romeo and Juliet becomes the dangerous love about which Friar Lawrence prophetically warns Romeo when he tells the passionate lover,

These violent delights have violent ends.
And in their triumph die, like fire and powder
Which as they kiss consume. (2.6.9-11)

So, while Juliet is cautionary and sensible initially in her approach to love, she later grows more impetuous in both her feelings and actions after secretly marrying Romeo.

Juliet consistently shows a maturity well beyond her age. Once she falls in love with Romeo, she becomes a determined woman willing to risk everything to follow her heart. In the famous balcony scene, she renounces her own family heritage if it means she can be with Romeo. She sees the foolishness of hatred because of a name alone. She can see past the feud.

Juliet remains loyal to Romeo, even after he kills Tybalt. Though for a moment she is conflicted about the death of her cousin, she soon quickly realizes the positive aspect of the situation:

My husband lives, that Tybalt would have slain,

And Tybalt's dead, that would have slain my husband.

All this is comfort (Act III, Scene 2, lines 105-107).

Juliet bravely accepts the Friar’s dangerous plan, facing her fears as she drinks the Friar’s potion. She would rather die than marry Paris, and she is ready to take her own life if the Friar’s plan does not work.

In the final scene, Juliet makes good on her promise. When she sees Romeo is dead, she kills herself because she refuses to live without him.

Juliet’s strong will, determination, and courage all point to the deep maturity she exhibits throughout the play.