I'm currently playing a Wizard in Nightmare, and I can confirm that there is some definite squishiness present. However, this is offset by the capability to do ridiculous damage to large numbers of enemies at a time.
If you're having trouble with large groups, consider what defenses you have. Is your armor spell always up? Keeping up Ice Armor can make a tremendous difference - when they hit you, they get frozen. Similarly, area-of-effect radius spells can be very effective - I'm still using Frost Nova at level 34. Wave of Force has a Stun ability at level 15, too. You'll get even better versions of these, though - Slow Time and Blizzard can be very effective slows, to bottleneck enemies so that you can deal some more damage and hopefully kill them before they get to you.
Early on, my typical strategy when surrounded was to Frost Nova, try to deal some damage, then when they were un-frozen, Wave of Force for the stun and either more damage or run away. The Evocation passive skill you get at level 13 is great for keeping these up.
As time passes, these situations will get a little easier to handle (though the enemies get much harder). Skills like Teleport and Mirror Image will come in very handy for those exact cases - keep an eye out for what level you unlock them. Teleport has some very good runes, and Mirror Image allows you to break conditions (like root, stun, and snare).
Honestly, much of what your aim is as a Wizard is to not even get to the point where there are enough mobs aroud you to worry. You may want to take another look at your current weapon situation - are you using the highest-damage weapon/off-hand combination you can find? For a good six levels I was using an enormous two-handed sword, because it gave +50 Intelligence (and brought my damage up significantly).
Don't give up Intelligence if you can bear it. Non-stat abilities (especially Magic Find) this early in the game are not your friend - you will be tempted to give up good equipment for them, and it isn't worth it. Intelligence will increase your damage and resistances, and should be well above all your other stats. I don't care if those rare boots have 9% Faster Run walk and 10% MF - you use those +20 Int boots over them, any day of the week. That being said, if it isn't a big damage hit, Vitality is also critically important for staying alive.
As a Wizard, you need the Templar. You need a tank. You've mentioned you're already using him, which is good. What abilities have you chosen for him? My recommendations are Heal, Loyalty, Charge, and Guardian (Guardian, in particular, is great for this situation, though obviously you don't get it until he's level 20).
Remember that followers get a multiplicative x2.5 bonus on stat bonuses from items. That means if you give them a +40 Vitality amulet, they'll actually get +100 Vitality (which is 1k life). Make sure you've got your follower properly equipped.