The question would be more correct if it were phrased the opposite way. Why did the Persians become Shi'a when most other Muslims (including most Eastern and Western Turks) are Sunni?

The answer is that Persia only became a Shi'a country fairly recently.

In 1501, Ismail of the Safavid Sufi order conquered the city of Tabriz, and then declared two things: that he was the Shah of Iran, and that the official religion of his empire would be Twelver Shi'a Islam.

Over the next decade, Ismail conquered most of the territory that had belonged to pre-Islamic Persian dynasties such as the Sassanians and the Parthians. In the regions he conquered, he and his descendants (the Safavids) took certain actions to convert his new subjects to Shi'a Islam.

Whether Safavid conversionary policies were coercive or not by the standards of their time is up for debate and I will not give an opinion on that.

Some actions they took included:

1. Desecrating the graves of important Sunnis such as that of the Sunni jurist Abu Hanifa
2. Embellishing the graves of important Shi'a figures in shrine cities such as Kufa, Karbala, Mashhad, and Qom
3. Removing state support from Sufi leaders and Sunni jurists
4. Inviting Shi'a clergy from other countries (notably Lebanon) to live in their empire
5. Having public festivals commemorating the martyrdom of the Shi'a Imams
6. Ritually cursing the names of historical figures viewed as usupers (Abu Bakr, Umar ibn al-Khattab, Uthman)

By the second century of Safavid rule, most of the Persian and Azeri populations had been converted to Shi'a Islam, as well as most of the Safavids' Arab subjects. Ethnic minorities in hard-to-reach or border regions of the empires often remained mostly Sunni, and remain so in the present day

In the Ottoman Empire, the rise of the Safavid Empire had ripple effects. Sufis and heterodox Muslims living in Asia Minor believed Shah Ismail was special, possibly even semi-divine. Some of them rose up in support of him. The Ottomans put down these rebels brutally, and began to crack down on heterodox forms of Islam

In spite of this, groups such as the Alevis in Turkey, the Alawites in Syria and more orthodox Shi'a in Lebanon persisted. While it is an oversimplification to call Alevis and Alawites Shi'as, they are definitely closer to Shi'as than Sunnis theologically

Sources:

1. Safavid Iran: Rebirth of a Persian Empire by Andrew Newman
2. Osman's Dream: The History of the Ottoman Empire by Caroline Finkel