It is not necessarily a run-on sentence; however, it is a sentence with a punctuation error. Our Rule 3b of Commas says, “In sentences where two independent clauses are joined by connectors such as and, or, but, etc., put a comma at the end of the first clause.”
Meat is healthy, but too much meat is not good for children.

Is the following sentence a run-on?

Meat is healthy but too much meat is not good for children.

The sentence is not a run-on sentence. In addition, the comma is optional. Our post Commas with Independent Clauses says, “Use a comma to separate two independent clauses joined by a coordinating conjunction—and, or, but, for, nor. You may omit the comma if the clauses are both short.”

I am reading and he is writing.

I know the folliwing is correct:
I am reading, and he is writing.

But, is it still correct when the comma is taken away?

Looking forward to your reply.

Thanks in advance.