In Turn 7 it's more likely the SMN will die than the Egi, since they're regularly targeted by a move called Circle of Flame which its for 2-2.5k damage. As long as the Egi is placed in the center of the map, they should only be hit by some AoEs and the floor killing mechanic, neither of which one shot pets. Turn 8 is almost impossible for your pet to die, since it should be stacked on the party, unless it's in front of the boss, which is most likely able to be one shot to death. I myself have limited experience in Turn 9, but I have not had any trouble keeping Garuda alive.

Keep in mind that Garuda takes roughly 80-90% reduced AoE damage, so what Circle of Flame hits for me, would do about 200-300 to her.