You cannot be dismounted from a stego using hardened plate mode, and hardened plate makes your stego take 50% already, so a stego is a great way to tame your first one, or for going into the redwoods in general. Be aware, the bleed damage is not reduced.
A good way I found to find these is to go to the redwoods riding a parasaur and walk into the forest spamming the parasaur’s ability. It will alert you to a thylacoleo in a tree if there is one, just make sure to look up at the trees near you if it finds enemies. Also would recommend bring a dino or two to protect you and your parasaur. Just signal them to passive when you find your thylacoleo. Hope this helps.
Try getting the health on your character up pretty high then fly around the redwoods (make sure the flyer is on passive) when a thylacoleo jumps at you start shooting at it as it usually can't run away. It doesn't take away too much health but I'd recommend to have a high amount just in case. This has always worked for me I'm not sure if it always works.
So. Taming a thylacoleo isnt really easy. Today was the first time i have tamed a thylacoleo and i must say, trying to find it is a pain. I used my quetzal and a parasaur called "bait" on turret mode to find him. If you cant use a parasaur try just going around the waters near redwoods and try to spot 1. Note: Better use a quetzal if you have one, dino's cant dmg you while holding them with quetzal. Find a thyla, shoot it with a single arrow and when ut falls go and grab it. Carry it into a cage or tranq it till it falls unconcious. You will need around 40-60 cooked meat depending on lvl (i am on 0.2 difficulty). Around 20-40 tranq arrows and at least 200 narcotics. When it falls asleep put in the food od choice and narcotics and wait for it to wake up. It takes about nearly an hour to tame. I hope this helps.It's torpor falls down really quickly so watch out for that. If you need help ask. My name is WakeyJakey