It's been awhile since I was playing as Christian factions in the desert, but here are some of the things I learned the hard way:
1. Your units are inferior in the desert, use strategy to win because it will be tough to win on the battlefield.
2. Naval superiority is a must--hit 'em where they ain't. Hit in rear regions so they have to come chase you down.
3. Strike many provinces at once by sea. If you launch 4 naval assaults, chances are that the AI will only be able to muster an overpowering army vs. 1 or 2. It's up to you as to whether to fight the larger forces, but you already have a couple of "easy wins", regardless.
4. Now that you have a few provinces under siege (hopefully with some fine generals trapped) you get to defend. This is far better than attack for the Western units, since the desert units must come to you now while your arbalests blast away at them. Standard techniques should work. Use arbalests to keep pesky horse archers and the like from running you ragged. You won't win by chasing horse archers, camels, desert archers, and Saharan cav around the desert.
5. Your *offensive* forces should be light. Old high valour units lacking much in the way of armour upgrades are the best in my experience. Anything with "white" armour tires pretty rapidly. I actually build light forces from recently acquired provinces with morale boosting buildings and low armour (usually "gray")--all just for desert attacks.
6. Light cav is good in the desert, but you need to kill the camels with archers/arbalests or spears before you can engage your light cav. Camels are hell on all kinds of horse cav. Fast units are good--because the desert units are fast and you don't want to have to charge just to keep up.
7. I rarely buy mercs. but that is one way to get an appropriate force. Bulgarian brigands are handy in the desert (fast). Militia seargants might be good--I've noticed the AI is very effective with them, but not sure how they do in the desert.
8. Attack and withdraw if needed to wear your enemy down. If you whip the first wave out of 3000 men, but your guys are all exhausted, it is probably best to withdraw from the field rather than face 2000 "fresh" peasants and desert archers. A high star general won't suffer much from a single orderly withdrawal, but losing him can be catastrophic (the "withdraw from battle" command is a lifesaver and the AI often uses it.) Afterall, you probably killed his best general or made a "good runner" of him. The next time you face the army it's strongest unis will be gone, and you will probably be able to sweep right over it.
9. Kill your prisoners unless you need the money. No sense giving them back their troops in a battle of attrition.