How is it possible that even after the existence of the third Hokage the fourth hokage emerges in the anime Naruto.
Answers
Regarding how they existed both as hokage, they didn’t, as there can be 1 hokage at a time.
I get your confusion, as up to this point in the anime/manga the hokage replaced the previous one after the death of the predecessor (2nd replaced 1st, and 3rd replaced 2nd).
However, in Hiruzen’s case, after he chose to end the 3rd Shinobi World War with a peace treaty, this enraged Danzo (Hiruzen’s old team mate, one of the village elders and a power house on his own) and this prompted Hiruzen to resign his position as Hokage as to his successor, Hiruzen initially wanted Orochimaru to be hokage, who was/is a genius but Hiruzen couldn’t convince him of what really matters (i.e. Konoha Village / will of fire), so he tried to convince Jiraiya to take the position, but Jiraiya thought he didn’t had it in him, being to carefree and not responsible enough.
So finally they settled with Jiraiya’s student Minato who became the 4th Hokage.
Hope this helps :)