You can find them very easily, in Scorched Earth. Just go near mountains, and look for the Thylacoleos with a Spyglass. Then, get up on a nearby rock. Try to shoot them, because unlike when you’re riding them, they don’t climb up to attack you. They will just stand at the bottom of the rock, so then you will be able to get many Tranq Arrows into him. (Crossbow is better than Bow.) After you tranq the Thylacoleo, you should place a Campfire and cook some meat, because Cooked Meat is actually better to use than Raw Meat. To craft a Thylacoleo Saddle, you need 20 Metal Bars, and around 250 Hide, and 200 Fiber, so you should keep an Argentavis with you, because they might Server very useful. Then, you just Tamed a Thylacoleo!
Best tactic I've come up with is to (double check before entering red woods) have your flyer following you and on Neutral/Attack your Target. When you're pounced off your mount, use a melee weapon in quick slot 1 (sword works perfectly) while your flyer backs you up in a few seconds, you'll almost certainly make it :P