Officially, never. The emperor was (and is) always considered the highest authority in Japan.

Practically, the question becomes whether you're asking about the person of the shōgun or the bakufu, the shogunate goverment. This is ignoring the Kenmu Restoration and accompanying Nanbokuchō, and the late Sengoku, when there was neither shōgun nor bakufu.

For the bakufu, the answer is 1221, following the Jōkyū War. Prior to it, the imperial court still had solid control around Kyōto and in the western provinces, and court-appointed rulers still had significant say over provincial politics. The war broke out as the imperial court tried to raise an army to crush the Bakufu. Instead, it backfired, and the imperial army was soundly defeated. The retired emperors and the emperor was exiled. Where-as before the war, the Bakufu assigned Kyōto Shugo to act as liason with the court and keep order in the capital, the office which was answerable to both the court and the bakufu. After the war, the office was replaced by the Rokuhara Tandai, which answered only to the bakufu, kept an eye on the court and all the lords of western Japan. The bakufu also used this chance to replace many of the lords of western Japan (not all) with its own officers.

However, by 1221, the shōgun himself was powerless. Ever since Minamoto Yoritomo, the first shōgun, died in 1199, the bakufu had been controlled by his in-laws, the Hōjō clan. It's not just that the Hōjō were regents until young children could come of age, which would happen many more times afterwards. It's that the Hōjō were so powerful they removed the second shōgun, who was soon assassinated. In 1219, the third shōgun was himself assassinated (though not by the Hōjō). The response of the Hōjō was to request someone of high family from the court (like maybe a member of the imperial family). This, and the bakufu's continuous encroachment on court authority, was what lead to war. Also notice there was no shōgun in 1221. After the war, the Hōjō got their wish, and the shōgun starting in 1226 until the end of the Kamakura Bakufu was essentially a powerless puppet, or worse a hostage from the court.

The first shōgun to actually control the bakufu while the bakufu controlled (more or less) the entire country was Ashikaga Yoshimitsu. As the third shōgun of the Muromachi Bakufu, he negotiated an end to the Nanbokuchō civil war in 1392, bringing the entire country under the bakufu's, and his, control.

Very interesting question, and concur with the sentiment given by u/ParallelPain . The nature of shogunal power is a very important condition to consider, when looking at Japanese history. Shogunates were central to the balance of power in Japan for the better part of around six and a half centuries (around 1185 - 1867). Within this period, there were 3 different shogunates (Kamakura, Muromachi, and Tokugawa shogunate). Throughout any of these given periods, the nature of power which the shogun had varied considerably. The question of “when did the Shogun become “supreme” does not have as straight forward of an answer as one might expect, because of the ebb and flow nature of authority which subsequent shoguns had within the different shogunates. One shogun might come close to what you would define as ‘supreme’, only for the next shogun to lose ground in this regard. To help give a better picture of this dynamic, u/morricane, myself, will explore the nature of shogunal power within the Kamakura, Muromachi shogunates respectively. Hopefully at somepoint, someone who is more familiar with the Tokugawa shogunate will be able to add on to this!