1) Keep in mind that you'll be hit by the AoE if you're in the red area when the cast bar fills up. There's usually a small delay between the time that the cast bar fills up, and the time that the mob does its attack animation. You can be halfway across the screen when the mob does its attack animation, but it won't matter. The game calculates your location when the cast bar fills up.

In other words, get out of the AoE as soon as it pops up. You may think you've got time to finish that cast, do one more attack, or check one more thing before moving. You don't. Because the attack actually hits before it appears to hit. Get out of the AoE right now. Your DPS/heals/enmity is going to drop to zero when you die. The most important thing is to get out of the AoE area right away.

2) I had the same problem with getting out of AoEs, especially on Titan. As others have suggested, the best method for me has been to step slightly out of the AoE area, and then use an ability. That forces the server to update your location. It's also much less stressful.

3) You also might want to check your ping to the FFXIV servers. If your ping is high, then the problem is your internet connection, and not the game. There are methods to lower your ping -- including using a VPN service or a ping minimizer -- but you'll have to google those, since I'm not going to endorse any of them.