I've fought them over and over and I continue to lose. My Sylestis are level 18 and level 15. I don't know what to do.

I think you should get up to level 25 first, try putting your Sylesti's on missions or grinding then reattempt it ^^

I would reccomend some good old level grinding. Defeat many, many enemies. I beat it with two level 20 (or around that) zolnixies. However, if it gets far too challenging with only that level, then go higher, attempt, and repeat that until you beat them. Also, afterwards, the boss of the next area was very difficult for me. And the one after that. So, this will become a strategy for the rest of the time as well.

Equipment.
You need Equipment on your pets. It's a vital part of pet setup.
It boosts all your stats, it boosts your damage.

At the moment between your Artemis and Rose Tyler (... an odd name choice for a male pet, but whatever) you have a total of one piece of equipment, and that's a level 0 common relic of the wrong damage type for the sylesti's elemental alignment (Phys damage on a fire element.)

(Also, neither light nor fire are typical 'Tank' elements (Light is fine, but usually the Tank is Water, Earth, or Shadow))

At the time I didn't know Rose Tyler was a guy so I gave her a girl name. And I bought equipment and some relics and they never showed up.

Also how do you take relics off and put equipment and relics on. It's been a while since I put the last relic on.

Go into your inventory, then click "go to equipment"
The equipment should be there.
Then scroll down to where it shows a pet. Then cick the little bar that says the pets name. Double click the equipment and click the appropriate spot on the bottom.
Repeat!

Hope this helps ;)

Thank you, I feel kind of stupid for not noticing that.

(Don't worry about it, I was having the same issue for some time as well.