By hitting Gyorg before dropping into the main room, you can skip the intro cutscene of the fight.
Use bow extension to shoot Gyorg from the spot shown in the video. This method is slightly slower than the weirdshot method optimally, and it also makes it harder (though not impossible) to hit Gyorg with the Zora barrier before the arrow stun wears off. A good visual cue for vertical aim is to align the bottom of the screen with the corner made by the wall and the floor.
These videos demonstrate using the faster method of bow extension and making it to Gyorg before the stun wears off, which is easier on English than on JP because of swimming control differences.
By damaging Gyorg next to the center platform, you are able to stunlock the first phase by swimming (with zora barrier) between Gyorg and the wall. The same carries into the 2nd phase, however now there are fish enemies swimming in the water as well. To stun Gyorg at the beginning of phase 2, you can use Zora boomerangs, a pot, the bow, or the hookshot.
This alternative uses Bremen Mask ISG and a bombchu to hover rather than a bomb. It is about 1 second slower than the bomb version shown above.
Hit Gyorg with the Zora barrier once so that one Deku Sword jumpslash will kill it. Then stun Gyorg near the platform, jumpslash into hops on the water to get ISG, and drown above Gyorg to finish the fight, skipping the mid-fight cutscene. For a method that doesn't get ISG, see the video below.
Using Fierce Deity, you are able to damage Gyorg from that platform with sword beam stabs. This is useful for SRM categories that have Fierce Deity early.
Standard methods for Remains Escape after defeating Gyorg are slow when you have lots of health because the best way to leave the dungeon is to void as Goron, play Song of Soaring to get to the entrance, and then leave via the turtle. This method lets you soar to your next destination right after voiding, which saves significant time if your next destination isn't Zora Cape.
Voiding late will mean you have to watch the Giant's cutscene, though Link will be Goron during the cutscene. Voiding early brings Link back to the room's entrance, where you can soar out as though you had done standard Remains Escape and voided.
Additional information regarding the Void Warp:
Gyorg's swallow works somewhat like a Redead grab. Link will be "freed" after you do 20 inputs, which can be a combination of A presses, B presses and control stick movement:
You can only avoid the damage if Link is grounded, as he must be able to roll. While he is grounded by default, if you press B followed by A while being Zora, you'll put Link in a swimming state, so you'd have to fix this by pressing B again. You can confirm that happened if the A button text changed from "Swim" to "Surface" upon pressing B.
Once Link is free, to avoid the damage you must overlap the roll's invincibility window (9 frames) with the moment the damage is dealt. You can either press A specifically on the frame right after pressing target (press target on frame 1, press A on frame 2) or, as human only, hold the direction that Link is facing and press A (spinning the control stick and mashing A may also work), just be wary of Gyorg's rotation affecting the direction you'll need to hold.
The time spent in Gyorg's mouth is not always constant, every A or B press while in its mouth reduces the time spent there by 2 frames, so it can take from 1.75 to 5 seconds from Link getting grabbed until he takes the damage. 1.75s requires pressing A or B on every frame, while 5s requires not pressing any of those buttons.
Interestingly, control stick movement frees Link without reducing the timer, so spinning the stick should be the most reliable way of avoiding damage.
Even you don't learn how to avoid the damage, you should at least mash A and B to escape the grab faster and save a couple of seconds.