Are you using the Standard Grid or the Expert Grid? How much HP on each character, and what are the enemies doing to you?

How do your stats compare to this video?
https://www.youtube.com/watch?v=TsDhq7wPwGs

Not having Cura means you're severely underleveled for Yuna, and this is often the case during the Bevelle split. Your primary form of healing at this point during the game should be using Use on Rikku and spamming Al Bhed Potions, which heal 1000 to all party members and can heal petrification. If Rikku is also underleveled, you're screwed.

smurf Kimahri

On the other hand, there's still hope, but this is the slow and difficult option. Unless the mobs on the bridge are actually killing everyone in your party in one hit, you can make your way through every battle one at a time. Simply doing "a lot" of damage means that you haven't died, which is good. If you have to use Escape to save even one of your members (or Tidus' Flee), you still can get some experience if you kill one enemy. Ability Distill also works on battles where you only kill one enemy and then leave.

Save after every battle, because it seems like you will need some serious RNG to generate the easy enemy formations. If you encounter any formation with the floating gun machina, just soft reset right then and there, because it sounds like you are just wasting time before you die. Fortunately, there is a formation that uses people with flamethrowers and the kicking bot. Burst down the dude, and then work on the machina. Unless the flamethrower dude is killing everyone every time he gets a turn, you should be okay.

Don't switch, don't get greedy trying to get experience on all six party members. You want these battles to end as fast as possible, so use Overdrives even while leveling (but don't use Auron's Shooting Star, because causing Eject will reduce experience gain). The moment you get even one character to survive enemy burst damage, leveling will become significantly easier. Set your Overdrive to the one where it charges on damage taken (Stoic or Comrade).

Other options include customizing armor and weapons. Fire Ward and Fireproof will make grinding flamethrowers possible. Defense +%, HP +%, SOS Protect can be very good grinding gear for cheap. Counterattack and Evade & Counter are great options to increase survivability and DPS. Brotherhood should have Waterstrike, which can help Tidus do more damage to the machina mobs.

If you really are having trouble with the mobs on the bridge, then you are simply not prepared for Seymour. The Highbridge is intentionally difficult. Your current "fast" option would be to try and get all of your Aeons to get full Overdrive bars and for Yuna to have a full bar so that she can Grand Summon. At this point, you spam Aeons and that should be the end of Seymour after some good luck.

You seem to have a major mental block with Yuna. Really, you don't need to get the abilities right now. It's more important to get the stats now, get past Seymour, and get to the next part of the story, where there are more manageable mobs. Macalania Forest has mobs that are from earlier in the story and the next zone, the Calm Lands, has mob formations that can be instantly be killed by Rikku/Steal.