Also you have to take in account that two times during the battle Pawniard got stuck and unable to defend itself and during these times it got hit with a direct Leaf Storm (which is a pretty powerful Grass-type attack. In game terms, it has 140 Power but STAB increases it to 210 Power. Even if you cut that in half that's still 105 damage) meaning it received all that damage.
As for Snivy being a Starter, remember it was a Wild Snivy so we don't really know how strong it is (it did defeat Trip's Servine in it's first battle against it). Plus Snivy has shown to be a pretty smart and competent Pokemon. It sees everything that's going on (thus explaining it's rivalry with Emolga who relies on getting away with tricking other Pokemon by everyone not paying attention and not seeing it) and goes into things with a plan (like when it tried to rescue Axew and Emolga from Simisear in DW025 "Emolga and the New Volt Switch!"). And if you want to go into game terms again, Snivy learns Leaf Storm at level 43 (though the anime tends to play around with how a Pokemon learns a move).
