I always pictured Link rewinding time, not transporting to a point in the past, creating an alternate timeline a
la Ocarina of Time.
To make it simplistic, I imagine multiple Links running around the current region they are trying to exorcise. For example, after Link depollutes the Southern Swamp, it is the night of the third day, and Link then travels to Snowhead. Link then rewinds time. It is now dawn of the first day, and while there is one Link in Deku form in his very first cycle, one Link travelling to Southern Swamp, there is now one Link in Snowhead, aiming to bring spring back again.
Also, at the same time, there is a Link rescuing Lulu's eggs and going through the water temple, as well as a Link exorcising the haunted Ikana, (it is this Link that will go on to face Majora immediatly after). These Links, though, are ones that the Snowhead Link is not aware of, as he didn't live through those points yet. He did, however, already defeat Odolwa, so he knows that there is another Link running around the swamps while he runs around the mountains.