The Samaritan woman was surprised that she would interact with a Jew (who usually do not get along well historically). In response, Jesus proclaimed how if the woman knew "the gift of God" as well as who he is, then the woman would ask for (and receive) living water (4:10). Essentially, Jesus is introducing the idea of getting "living water".
However, when the Samaritan woman is thinking of "living water", she is probably thinking of the drinking water like in the above picture. That gave her a few questions. Jesus had no water tools, and the Samaritans knew that Jacob's well is important in her life. Because of how central Jacob's well is to drawing water, it is hard for the Samaritan to imagine some person greater than Jacob, who made the well in the first place.
In response, Jesus made a contrast between the water of Jacob's well and "living water". Drinking from the water of Jacob's well does help with hydration but is not permanent (4:13). Anyone who does athletics knows that one drink of water isn't enough. On the other hand, "living water" is more permanent because it creates a "fountain of water" that leads to "everlasting life" (4:14). Jesus's use of "living water" is actually figurative language (especially because of the "living" that makes it different from other water). As we'll see later (and from John 3), "living water" is symbolic of renewal that Jesus provides that results in everlasting life.
In the conversation, Jesus made a transition from water to "living water". Such transitions make sharing God's good news about Jesus smoother.