Well, first thing's first: the tank is at fault for causing these problems. As you said -- that many mobs on him, and at 1/4 HP by the time you can catch up and cast anything? That's a joke. No matter how good a healer you are, there's little you can do at that point, especially if he pulls this stunt repeatedly and your cooldowns are blown. From his behaviour, I'll bet he really wants to force a speedrun, and is just using high damage skills rather than defensive abilities.
Second, if you've been healing without regular problems for this long, chances are you are not doing anything fundamentally terrible.
Third, he says you're doing your "rotation" wrong? Baloney, there is no such thing as a healing rotation.
As far as tips, I'd say:
* Regen the tank at the start of a fight, and keep it on at all times. I ease off the Regen if there's one mob left and a fresh cast will persist long after it dies, just because if it's still on when a new group of mobs is encountered it seems to initially draw aggro to me.
* Don't be afraid to use Shroud of Saints often, in bosses or longer fights. It is not an emergency "Low MP" button, but rather something you should be using regularly to avoid having low MP.
* Contrary to what that one person told you, don't put Cleric Stance on. You are a healer, not a DPS. By all means, if everyone's relatively topped up and there are no imminent emergencies, throw it on for a moment and dish out some damage. But that is an extra bonus on top of your regular role, only if the run is a cakewalk.
* Eye for an Eye is an excellent Arcanist cross-class ability. Buff the tank with it, and any mobs that hit him will have their damage output decreased. Naturally, this is more useful the more mobs are whacking on the tank. If there's a big pull coming up, throw it on. I use it all the time, it's great.
* To be perfectly honest, I use regular Cure the majority of the time, and only switch to Cure II when things are getting hairier and the damage is outpacing the heals. Maybe Cure III if it's really bad. As others mentioned, Divine Seal and Presence of Mind are very helpful in a pinch (though I am bad at remembering to use them).
* Are your bonus attributes from leveling put into Mind? Because that's where they belong.
* I used to put Stoneskin on the tank before a lot of fights, but I'm doing it less and less now because I find I'm just standing there watching him, waiting for damage to start registering. Seems kind of pointless. Instead, I put it on me and the more squishy DPS, to lessen the impact of somebody suffering a stray attack or AoE.
* Not sure if this depends on your connection or not, but when I want to cast stuff repeatedly, I don't wait for the cast bar to disappear and the spell to connect. I hit the next spell slightly before the previous cast bar is full, so there is no interruption or downtime.
* A bit hypocritical of me to say, since I haven't actually gotten around to learning it yet, but apparently Swiftcast from THM is a must-have. If somebody does go down in a tough fight, Swiftcast + Raise will make a world of difference.