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On the one side of the family, we have the union between Tamar and Yehudah, which, while technically a fulfillment of Yibum, Yehudah thought she was a harlot. From this union came Peretz (Bereishis 38:15-29), the forebear of David HaMelech (Rus 4:18-22).

On the other side of the family, we have Rus, a descendant of Moav. He was born from the union of Lot with his eldest daughter (Bereishis 19:36-37).

Continuing down the line, David himself married Bas Sheva, after seeing her beauty and sending her husband Uziah to the front lines of the war to be killed in battle such that she would be single (Shmuel 2:11:2-27). From this union came Shlomo HaMelech (ibid. 12:24).

Shlomo himself married Ne’amah, a descendant of Ben-Ami, the son of Lot with his other daughter (Bereishis 19:36,38). Ne’amah went on to be the mother of Rechavam (Melachim 1:14:21).

All three of these incidents were either technically permitted or at least done for noble causes. David married a single woman, after her husband was killed as rebelling against the king (Kiddushin 43a). Yehudah, even if he didn’t know it, was fulfilling the mitzvah of Yibum. And Lot’s daughters are praised for their actions, as they thought that the world had been destroyed and it was up to them to continue the human race (Horayos 10b-11a). Yet, they all smack of something illicit: David was reprimanded for this incident (Shmuel ibid. ch. 12). Lot’s daughters’ actions are termed a “sin for Heaven’s sake” - but still a sin. And while Yehudah was never punished for this, he did have the intimacy under the pretense that she was a harlot.

Why must the Davidic dynasty come from such a background? Why couldn’t it be a normal family with normal marriages?

See the gemara in Yoma 22b.

This was orchestrated in order to ensure the longevity of his kingdom.

As Rabbi Yoḥanan said in the name of Rabbi Shimon ben Yehotzadak: One appoints a leader over the community only if he has a box full of creeping animals hanging behind him, i.e., he has something inappropriate in his ancestry that preceded him. Why is that? It is so that if he exhibits a haughty attitude toward the community, one can say to him: Turnand look behind you and be reminded of your humble roots.

Sefaria's explanation based on Rashi spells out:

This is why David’s kingdom lasted while Saul’s did not, as David descended from a family with problematic ancestry, namely Tamar (see Genesis, chapter 38) and Ruth the Moabite (see Ruth 4:18–22).

Good question.

(1) We each have two parents, four grandparents, eight great-grandparents, and the numbers keep growing exponentially in powers of two. So each one of us has A LOT of ancestors – thousands upon thousands. The chances are overwhelming that some of these people did not rank high on the scale of virtue. In fact, the Talmud tells us: Why was humanity derived from a single couple, Adam and Eve? So that no one should be able to tell his neighbor, "My ancestors were better than your ancestors". [Sanhedrin 37a] So, before you ask, “Why is God making the Messiah come from such dubious ancestry?”, ask yourself, “Does God really have a choice?”.

(2) The 19th century Hasidic master Rabbi Zadok HaKohen of Lublin writes, in his book Tzidkat Ha-Tzaddik:

“The redemption will emerge precisely from a place of lust and sin, by means of repentance (teshuvah). The Talmud says, in Sanhedrin 98a, “The son of David [i.e., the Messiah] will come only in a generation that is entirely wicked.” David is the archetype for the messianic soul because he showed how to make repentance into a life principle… And just that is the realization of ultimate fulfillment – that the root of evil will be transformed to good… At that time, the lowest will become the highest.”

(Actually, a direct check show that Sanhedrin 98a says: “Rabbi Yohanan also said: The son of David will come only in a generation that is entirely wicked or entirely righteous.”)