You play Shyv to straight up beat people in raw stats. She has mixed damage which makes her hard to itemize against, she has built in resistances so she's naturally tough even without extreme amounts of investment, and depending on what you build she is impossible for a lone person to kite.
Don't gank pre-6. If you're an experienced Jungle player in general, pay attention to the enemy Jungler and laners so you can countergank where necessary. As mentioned, Shyvana stat checks people really well, meaning that in a 2v2 brawl drawn out by your arrival, you can easily turn around an enemy gank.
Counter jungle just for it. If you have the right runes and masteries, you are going to burn through the Jungle (pun intended) at stupid speeds. Even with just Blasting Smite (no enchantment, just
and literally nothing else), I can push Shyvana to clear all 4 non-buff Jungle camps on my side of the map, kill the Scuttle Crab, and make it back to my first non-buff Jungle camp before it has even respawned. Instead of wasting downtime by waiting for your camps to respawn since you clear so fricking fast, go ahead and Counter Jungle whenever it is safe to because all it does is give you less downtime as you accumulate gold and experience. This is key to making Shyvana the stat checking monster she is by mid to late game.
If your team loses 4v4s, or hell even 4v3s, you probably aren't going to win the game as a whole anyway. Regardless, split pushing is one of the better ways of highlighting Shyvana's strengths if you're facing a team with a lot of peeling potential, so you fight just have to suck in your gut and deal with it.
I actually am having stupid fun with a Splitpush Shyvana build that I've been messing around with. Only 8 games with it so far but a 75% win ratio is just nice.