After all, reward and punishment are given in response to actions done through free choice. It doesn't seem logical to punish a person just because his parents sinned when he himself did nothing wrong. Even so, there are cases where God punishes a person by having his descendants suffer.

Bring examples and we can discuss it.

"The fathers have eaten sour grapes, and the children's teeth are set on edge."

https://he.m.wikipedia.org/wiki/%D7%90%D7%91%D7%95%D7%AA_%D7%90%D7%9B%D7%9C%D7%95_%D7%91%D7%95%D7%A1%D7%A8_%D7%95%D7%A9%D7%99%D7%A0%D7%99_%D7%91%D7%A0%D7%99%D7%9D_%D7%AA%D7%A7%D7%94%D7%99%D7%A0%D7%94

"In those days they shall no longer say: 'The fathers have eaten sour grapes, and the children's teeth are set on edge.' But each man shall die for his own iniquity; every person who eats sour grapes, his own teeth shall be set on edge."

See also the whole of chapter 18 in Ezekiel.

What about the fact that the son of David and Bathsheba died?

A child's death is a punishment for them, not for him. Except that there it is described that the child also suffered from it for a week. That indeed requires further examination, especially in light of the sources above that children are not punished for their parents' sin (the law of a mamzer is also difficult in this respect).
Maybe there was some special need here for such a punishment for the parents, even if it came at the child's expense. It's like punishing a sinner with death and his family suffers from that. Or just suffering that comes to a person without his having sinned because there is some need for it. This requires further examination.

So in summary, intergenerational punishment is not really something that exists?

As I understand it, it should not exist, and the verses also say that it does not. In practice, perhaps there can be situations where this is done if there are good reasons for it (that is, goals that cannot be achieved otherwise).

Got it. Thank you very much!