The story of the origination of Gautami Ganga is written in Varaha Purana, adhaya 71 as:
Gautama muni asked Lord Shiva for Ganga. Once he received Ganga, Gautama muni poured ganga jal on top of a dead cow and the cow arose. The place where this event took place is called Gautami Ganga or Godavari Ganga.
Matsya Purana, adhaya 22, shloka 58-59:
In Godavari river, our ancestors are known as Apsaroyuga.
- Apsaroyuga is known as Apsaro, which flows from Kal (Shiva) Pokhari towards Gautamitat (intersection of Barun and Arun river). There is no river that flows towards Gautamitat coming from current Godavari.
Brahma Purana, adhaya 78, shloka 77:
Godavari and Bhagirathi originate south and north respectively in Bindagiri from the same mountain.
- The current Bhagirathi originates in Himalayan range and Godavari is in Maharashtra. They both do not form from the same mountain.
Brahma Purana, adhaya 89, shloka 38-45:
Arun river (known as Tapi in the shloka) and Yamuna (Barun river) intersect at a place called Gautami Ganga (Gautamitat).
Our current Godavari river is in Maharashtra. But this doesn't match with what the shastras tell us. So where is Godavari?
Note: I do not wish to offend anyone regarding my question. Our current thirthas are as holy regardless of their authenticity and I respect their location. Being a Hindu, I would never disrespect any thirtha just because they do not match with shastras. Many thirthas have changed locations and this is why I try to find their original location. Sorry if I have offended anybody.