It's a bit complicated, and Persia isn't really my focus/expertise but i'll try to answer your question, sorry if it doesn't satisfy you

Now i think you're a bit confused there, Safi al-Din established the Safawi sufi order, not the dynasty. He also doesn't immediately converted to Shi'ism. With that said, a bit of background

Shaykh Safi al-Din was probably of Kurdish ancestry. His ancestors seem to have lived in Azerbaijan since the 11th century. He attached himself to Shaykh Zahid Gilani, the head of a Sunni sufi organization in Talish. After Shaykh Zahid's death, Shaykh Safi assumed the headship of the organization and moved to Ardabil, where he lived for the rest of his life. He was also a popular figure in the Ilkhanate, and he was held in great respect by the Mongol converts who preferred Sufism than the "orthodox" Islam of the ulema. His biography, the Safwat al-Safa contains a wealth of anecdotes about Shaykh Safi’s activities and miraculous deeds. Whether the miraculous claims are true or not, this book proves that he was, rather undoubtedly of a Sunni persuasion. This later becomes a problem to his Shi'i descendants after they had seized the throne of Persia. In the 16th century revised versions of the book were produced in which Shaykh Safi is represented as having professed more suitable views (fortunately, not all the manuscripts of the authentic version were destroyed). After his death, his order has been a highly successful as a more or less conventional Sunni sufi organization, gradually acquiring disciples and property. Its leaders had shown little sign of political involvement in the tangled affairs of the Turkmen era, however after the death of the fourth leader Shaykh Ibrahim things begin to change

Ibrahim’s successor was his son Junayd, but the actual head of the order at Ardabil throughout Junayd’s life was his uncle Ja'far. Ja'far had been installed with Qara-Qoyunlu support and Junayd was expelled from Ardabil, and obliged to go on his travels. It was this period, that was crucial in the transformation of the Safawid order from a peaceful and essentially “apolitical” sufi organization to a militant, though not yet necessarily a Shi'i, political movement. Junayd spent his long years of exile initially in eastern Anatolia, but later in northern Syria. There he recruited considerable numbers of devoted followers from the Turkmen tribesmen of those regions. These Turkmens are devoted to Junayd, and with that an extremist religious beliefs based on the Safawi context started to emerge, which regarded not only Ali, the first Shi'i imam but also Junayd is of divine nature. However, this isn't Shi'ism yet, as the movement only contained some elements of Shi'ism. With his loyal followers, Junayd allied himself with Uzun Hasan of Aq-Qoyunlu who gave him his sister in marriage, thus cementing relationship between them. Because this group viewed themselves as ghazis, they started their militant career by conducting raids into nearby Christian kingdom of the Caucasus. Unfortunately, in order to reach the kingdoms they had to cross the territory of the Muslim Sirwan-shah, who regarded them as enemy and attacked them, killing Junayd. His son Haydar was brought up at Uzun Hasan’s court and married to his daughter, thus strengthening the family relationship between the two houses. After going to Ardabil, Haydar was said to have invented the taj, a red twelve-gored hat for his followers to wear seemingly in honour of the Shi'i imams (That's why Turks called them the Qizilbash, meaning red-heads). However, their movement was still not Shi'i yet as Haydar isn't influenced by Shi'ism and respect for the imams did not necessarily mean Shi'i allegiance.

However, with Uzun Hasan's death, his successor Yaqub regarded Haydar and his militant forces as potential danger. Haydar then mounted a raid into the Caucasus emulating his father, but again was attacked and killed fighting Shirwan-shah, who this time was supported by Yaqub. Haydar's son Sultan Ali succeeded him as the head of the Safawi order, but he and his brothers were imprisoned by Yaqub in the Aq-Qoyunlu fortress of Istakhr in Fars. Sultan Ali was then killed, but his younger brother Ismail (aged only seven) escaped and assumed leadership of the order. He found refuge in Gilan, where he was educated by a tutor who was also presumably a Shi'i of some sort. Thus by this time, Ismail who has been thoroughly educated and exposed to Shi'i influence started to reform his order according to Shi'ism, though not an orthodox kind. After he completed his study, the twelve-year old Ismail rode out of Gilan in order to have revenge on both Shirwan-shah and Aq-Qoyunlu. His religious appeal of a divine ruler, his charisma, and his rather relaxed Safawi movement compared to the rigid and bureaucratic Ottoman system of government and taxation appealed to a lot of Turkmen nomads. With that, he defeated the Aq-Qoyunlu which was divided by factionalism and seized the throne as the divine leader of a Turkmen religious movement. He himself later imposed Twelver Shi'ism belief that's different from his original creed, but that's probably a thing that people better than me on this matter should tell.

Bibliography

Medieval Persia, 1040-1797 by David Morgan

Safavid Iran, Rebirth of a New Persian Empire by Andrew J Newman