While leveling up, standard and crafted gear (even high-quality crafted gear) has the same Item Level as it does equippable Level, so with those you can't get above your character level, while if (for example) a particular slot gets a better piece available each four levels (which is fairly typical) that's three-fourths of the time that it's Item Level would be below your character level, even if you get every upgrade as soon as you're able to equip it.

Dungeon drops (and a few other specialized pieces) can have higher Item Levels than their equippable level, but that only lets you get above your level if you get lucky enough to get the drop immediately, before you've leveled up past its Item Level.

So, typical leveling gear might have one or (if you're particularly lucky) two pieces with an Item Level a level or two above your level, a few with Item Levels equal to your level, and several with item levels below your level, simply because nothing better is going to be available for a while. (Accessories in particular may sometimes be quite a few levels lower, since those don't get new upgrades available nearly as often as armor or weapons.) When these Item Levels get averaged together to form your overall Item Level, even good gear you've been upgrading whenever available will still end up being lower than your character's level most of the time.

This changes as you near endgame. Endgame crafted gear is no longer limited to the Level = Item Level rule that nearly all lower level crafted gear follows. And dungeon drops or token traded gear can get item levels that are drastically above their equippable level (as opposed to perhaps a couple levels above as the good leveling dungeon drops get). (I believe this starts at around level 45, though the possibilities expand tremendously at 50.) So at that point is when people's average ILevel can shoot up well above their actual level.