Disclaimer: This route has only been completed on the PC version, but all the tech should work on console as well. The only version I suspect will have differences is the Switch version, given it's "downgraded" performance. It will be helpful to watch the WR video as well. Tag me on discord if you need further help.

Terms:

Potentially useful terminology:
EF - Edge Fortress - One of the maps in the game, part of the loop, the place you spawn on new game
Purgatory - Place you spawn after you die
Waterfall - One of the maps in the game, part of the loop.
Darkness - One of the maps in the game, part of the loop. 
WF - White Fountain - Ending location for the run
P - Photomode - Useful for glitching and falling safely
Teleport - General movement mechanic of the game
The Icelands - The are where you can get more life after reaching permadeath state
Fountains - Map where the end goal of the game is. 
Double map load - Having two maps loaded at the same time. Can be done with Purgatory or transition jank on one transition. 
Triple map load - Double, but thrice!
Transition jank - Sort of an umbrella term for glitches involving going through a transition and immediately teleporting to the previous map. Done by walking backwards into a transition and releasing a teleport as the map change happens. Poor name choice. 
Long dash - Upgrade in the game that you usually get later in the game. Unused in this run.

Guide:

After starting a new game, immediately walk off the edge of the pillar to die.

Once you're in Purgatory, make your way to the spiral staircase basement in the middle. Once entered the bottom "donut" room, start loading a teleport to the other side of the transition. Make sure you do this from inside the room. Release the teleport and immediately press P (enter photomode). If you time your pause right, you'll get the photomode pause before you enter The Icelands. Assuming you got this, start exiting photomode, but keep pressing P as this extends the state of exiting photomode. This is required for setting up the glitch to load the transition to Fountains in The Icelands. Once the transition happens and you see the black lines indicating that the transition is blocked goes away, you can safely exit photomode.

After entering The Icelands, we prompty return through the transition again and go up one level of the spiral staircase. There's an enemy in the room a room we need to teleport into. It's on the left of the level we just went up to. Teleport in and walk through the enemy. You will not die. Then walk up the rest of the staircase and walk to the water below the closed gate to Fountains. After this, make your way to The Icelands again. If everything was done correctly, you should see the transition to Fountains in The Icelands.

Make your way over the ice to the transition we just glitched into view. You can teleport up to it. Doesn't matter which side. Walk through the transition so you have The Icelands and Fountains fully loaded. Do two transition janks from the Fountains side of the transition. You should bounce when doing it to get the best result. You'll hear the sound effect of going through the transition playing twice. Based on timing, you can get it a bit differently, but this one is a safe bet, do this one twice. This should give you a triple map load of Fountains, Purgatory and The Icelands. Make your way across the ice to the invisible wall in Purgatory and walk into the water.

If you have triple map load while in Purgatory, make your way down the spiral staircase again, but this time go around the donut on the right side so you enter Purgatory again, this makes The Icelands map go away as you technically exit it into Purgatory. This reduces down to a double map load again. Now just make your way up and resurrect, and you should keep Fountains with you as a double map load while alive, and you're back in Edge Fortress.

Once back, make your way up the tower and perform a tower jump down to the tower we see across the water from the starting point (Will expand on this, but use the WR video as reference for now). You normally need a long dash to get there. Do a photomode teleport so you don't die from the fall. It's a bit precise. If done correctly, make your way to the Waterfall map.

Once in the Waterfall map, make your way across the water with photomode teleports onto the water. This is a bit precise because you might clip the water before entering photomode. If done successfully, you can enter the cave to where the transition to Darkness is. Do a transition jank from the Waterfall side, or do a wall flick towards the Waterfall cave entrance after going through the transition. By the cave entrance, you should then be able to fall off the map directly into White Fountain. Use photomode to descend safely.