So I started a new character yesterday. It is an archer build and it seems to be leveling up really slowly. I was wondering if archers just level up slower than other fighting styles or if it is just me. I have been playing for 19 hours and am only LVL10.

Use Thief Stone. Not Warrior Stone. This is very common mistake.

Fewer sneak shots. A lot of enemies that you can one-shot kill are still easy to kill if you do not sneak. Not sneaking means you are shooting two or three shot to kill someone, instead of just one.

Use the best bow type you can find. Carry a high quality unsmithed bow.

Bound bow is best for leveling up quickly. You can find Bound Bow book at Fort Amol.

If you are okay with receiving training, you can recruit Faendal from Riverwood and receive free training to Level 50.

Hunting is very good for leveling up archery. Hunting mammoths is especially good. They can take a lot of hits before dying. Hunting Trolls is also very good. Don't try to kill them quickly with sneak shots. Find a safe perch, stand up, and kill them slowly with many shots from your best unsmithed bow. Use cheapest/weakest arrow.

If your killing things with fewer hits then with a melee weapon then I guess that could be the reason. I've noticed on my Mage destruction levels slower then my conjuration even though I use destruction more.

To level destruction quickly, use your most expensive spells. Incinerate is very good for leveling, then Fire Rune, then Firebolt, then Flames. Fireball is a great spell but terrible for leveling up.

This sounds like very good advice, I almost forgot about the thief stone... just outside Riverwood I believe.