How did Shigeru (Gary) defeat Natsume (Sabrina)?

[Fantastische Hure]

Well, maybe if he defeated her after Satoshi (Ash) it'd be easier to imagine but find that hard to imagine.

 

[Papasmurf]

gary oak 10 badges - Google Search

He had different sets of badges after the first 5

 

[Yoshi]

He most likely fought her after he fought Erika and Koga like you're actually supposed to do in the games, so he got strong enough to defeat her without it feeling forced.