....in the third leg of a twelve-hour marathon of the previous season's America's Next Top Model....
I tried my level best to get out of Support Group while sitting on the couch with my mom in the third leg of a twelve-hour marathon of the previous season's America's Next Top Model....

From the book, The Fault In Our Stars.

What does "the third leg" mean here?
Can you help me?
It certainly does not refer to men's sexual organ.

Question

....in the third leg of a twelve-hour marathon of the previous season's America's Next Top Model....
I tried my level best to get out of Support Group while sitting on the couch with my mom in the third leg of a twelve-hour marathon of the previous season's America's Next Top Model....

From the book, The Fault In Our Stars.

What does "the third leg" mean here?
Can you help me?
It certainly does not refer to men's sexual organ.

Answer

https://www.macmillandictionary.com/dictionary/british/leg_1#leg_1__17
https://www.macmillandictionary.com/dictionary/american/leg_1#leg_1__17

Answer

In the third section or part of the marathon session 
It's an expression coming from relay races I think where each section/part is referred to as a leg.