More

santosh121The more was I reading the tragic story, the more tears were coming down my cheeks. Is the sentence correct?

I think now it's OK.  ....

I don't think inversion is usual in 'the more..., the more' constructions.

There are two other points:

1. the failure to say the more the tears... invalidates the construction, in my view.
2. it's odd to use a continuous tense in such a construction.

I'd change it to The more I read this tragic story, the more the tears ran down my cheeks.

Thanks to you, Thomas, for your corrections. I didn't know it was odd to use continuous tenses in such constructions. 

Please tell me if it would be odd to say these sentences, too. 

- The more she's practicing swimming, the more she's becoming adept at it. 
- The more they were talking to each other, the more they were getting familiar with each other. 

Thank you on

santosh121 The more was I reading the tragic story, the more tears coming down my cheeks. Is the sentence correct?

It's not correct because the second part is missing a finite verb. You need something like this:

The longer I read the story, the more the tears came down my cheeks.

CJ

TomJPlease tell me if it would be odd to say these sentences

I'm not Thomas, but I'll put in my two cents anyway.

You need:

The more she practices swimming, the more adept she becomes at it.
The more they talked to each other, the more familiar they got with each other.

I agree that continuous tenses in the more-

I can't agree more with Mr. Tompion. Alternatively, Id break the sentence up for better flow you know? like:

"I was reading (whatever). The more I read, the more I cried."
or
"I was reading.. . The words kept collecting into tears until they brimmed and flowed down my cheeks."

but to address your premise,

"The more I read the tragic story, the more tears flowed d