David stood on the platform

[Bassim]

Are my sentences grammatically correct?

David stood on the platform, waiting for his train when he saw a pickpocket removing a wallet from an old woman's bag. "Hey, what are you doing?" David shouted, but the man ran away. David ran after him, but his large body made him slow down and gasp for breath.

 

[emsr2d2]

Are my sentences grammatically correct?

David [STRIKE]stood[/STRIKE] was standing on the platform, waiting for his train, when he saw a pickpocket removing a wallet from an old woman's bag. "Hey, what are you doing?", David shouted, but the man ran way. David ran after him, but his large [STRIKE]body[/STRIKE] frame [STRIKE]made[/STRIKE] forced him to slow down and gasp for breath.

See above. The two things (standing on the platform and waiting for the train) were happening at the same time so I would use the past continuous. Even though "made" means "forced" here, I find "forced him to" more natural. I honestly can't explain why.

Some people with large bodies/frames are able to run quite fast. Did you mean that he was too fat to run after the thief (but you were being polite)? I would probably use "thief" rather than "pickpocket" but that's only because he didn't actually take the wallet from a pocket. Also, most women carry a purse (BrE) and men carry a wallet. It's not impossible for the woman to have been carrying a wallet; it's just less likely.
A purse in AmE is a handbag in BrE.

 

[Bassim]

You are right emsr2d,
I've used "large body" meaning fat. Should I write instead:
"but his fat body forced him to slow down..."?

 

[emsr2d2]

You are right emsr2d2, [STRIKE]I've[/STRIKE] ​I used "large body" meaning fat. Should I write instead:
"but his fat body forced him to slow down..."?

I'd probably say something like "but, because he was so fat/because of his weight, he quickly found himself gasping for breath and was forced to slow down."

"Because of his weight" would be taken to mean that he was carrying excess weight.

 

[Roman55]

but the man ran way.

I assume this is a typo. You obviously meant away.