It is due to Saraswati's curse that Ganga is born as a river on Earth and became a sin destroying river.
In Chapter 6 of 9th Book of Devi Puraanam, Naarada asks Rishi Naaraayana- Why did Ganga curse Saraswati to become a river? Why was there a quarrel between Ganga and Saraswati?
Then Naaraayana answers Naarada-
Reason of fight
Devi Puraanam, Book 9, Chapter 6
शृणु नारद वक्ष्यामि कथामेतां पुरातनीम् ।
यस्याः श्रवणमात्रेण सर्वपापात्प्रमुच्यते ॥ १६ ॥
लक्ष्मीः सरस्वती गङ्गा तिस्रो भार्या हरेरपि ।
प्रेम्णा समास्तास्तिष्ठन्ति सततं हरिसन्निधौ ॥ १७ ॥
चकार सैकदा गङ्गा विष्णोर्मुखनिरीक्षणम् ।
सस्मिता च सकामा च सकटाक्षं पुनः पुनः ॥ १८ ॥
विभुर्जहास तद्वक्त्रं निरीक्ष्य च क्षणं तदा ।
क्षमां चकार तद्दृष्ट्वा लक्ष्मीर्नैव सरस्वती ॥ १९ ॥
बोधयामास पद्मा तां सत्त्वरूपा च सस्मिता ।
क्रोधाविष्टा च सा वाणी न च शान्ता बभूव ह ॥ २० ॥
O Naarada! listen I am reciting the ancient tale, hearing which, one gets free from all sins. 16.
Lakshmi, Saraswati and Ganga are three beloved wives of Vishnu, and they live together with harmony always close to Hari. 17.
But one day, Ganga, smiling and full of passion, started looking at Vishnu's face and drew continuous side glances on Him. 18.
Vishnu also smiled looking at Her face for moment. Seeing all this, Lakshmi took no offense, but not Saraswati. 19.
Then Lakshmi, who is Sattva incarnate, smilingly consoled Saraswati. But filled with anger Saraswati did not become quiet at all. 20.
सर्वत्र समताबुद्धिः सद्भर्तुः कामिनीं प्रति ।
धर्मिष्ठस्य वरिष्ठस्य विपरीता खलस्य च ॥ २२ ॥
ज्ञातं सौभाग्यमधिकं गङ्गायां ते गदाधर ।
कमलायां च तत्तुल्यं न च किञ्चिन्मयि प्रभो ॥ २३ ॥ गङ्गायाः पद्मया सार्धं प्रीतिश्चास्ति सुसम्मता ।
क्षमां चकार तेनेदं विपरीतं हरिप्रिया ॥ २४ ॥
A good husband, who is religious and well qualified, always have equal liking of all his wives. Only a stupid acts contrary to this. 22.
O Vishnu, holder of Gadaa! Your liking to Ganga is clearly visible, and You have same love for Lakshmi too. Only I am deprived of your love. 23.
This is why, Ganga and Lakshmi like each other. And thus, Lakshmi does not do things against this harmony between them both.
The fight and curse of Saraswati to Ganga
सरस्वतीवचः श्रुत्वा दृष्ट्वा तां कोपसंयुताम् ।
मनसा च समालोच्य स जगाम बहिः सभाम् ॥ २७ ॥
गते नारायणे गङ्गामुवाच निर्भयं रुषा ।
वागधिष्ठातृदेवी सा वाक्यं श्रवणदुष्करम् ॥ २८ ॥
हे निर्लज्जे हे सकामे स्वामिगर्वं करोषि किम् ।
अधिकं स्वामिसौभाग्यं विज्ञापयितुमिच्छसि ॥ २९ ॥
मानचूर्णं करिष्यामि तवाद्य हरिसन्निधौ ।
किं करिष्यति ते कान्तो ममैवं कान्तवल्लभे ॥ ३० ॥
Hearing Saraswati's words and seeing Her full of wrath, Hari thought for a moment and went out of the court. 27.
Narayana gone out, Saraswati, who is main deity of speech, spoke full of anger and fearless to Ganga, things which are hard to hear. 28.
O passionate and shameless Gangaa! being elated with pride of your husband's love, now you are wanting to show that your husband loves you more. 28.
Let me crush down your this pride here in front of Hari. O beloved of husband! let us see, what your husband do to me.
इत्येवमुक्त्वा गङ्गायाः केशं ग्रहीतुमुद्यता ।
वारयामास तां पद्मा मध्यदेशं समाश्रिता ॥ ३१ ॥
शशाप वाणी तां पद्मां महाबलवती सती ।
वृक्षरूपा सरिद्रूपा भविष्यसि न संशयः ॥ ३२ ॥
विपरीतं ततो दृष्ट्वा किञ्चिन्नो वक्तुमर्हसि ।
सन्तिष्ठति सभामध्ये यथा वृक्षो यथा सरित् ॥ ३३ ॥
Saying this, Saraswati moved forth to catch hold of Gangaa's hair, but Lakshmi, standing between them, caught hold of Saraswati's hand. 31.
Then Vaani(Saraswati), who is pure and extremely powerful, cursed Padmaa(Lakshmi) that you may become a tree and a river without no doubt. 32.
Even after seeing Gangaa, if you Lakshmi, standing between us, cannot speak anything just like a tree or a river, then let you become them both. 33.
त्वमुत्सृज महोग्रां च पद्मे किं मे करिष्यति ।
दुःशीला मुखरा नष्टा नित्यं वाचालरूपिणी ॥ ३६ ॥
वागधिष्ठात्री देवीयं सततं कलहप्रिया ।
यावती योग्यता चास्या यावती शक्तिरेव च ॥ ३७ ॥
तथा करोतु वादं च मया सार्धं च दुर्मुखी ।
स्वबलं यन्मम बलं विज्ञापयितुमिच्छति ॥ ३८ ॥
जानन्तु सर्वे ह्युभयोः प्रभावं विक्रमं सति ।
इत्येवमुक्त्वा सा देवी वाण्यै शापं ददाविति ॥ ३९ ॥
सरिक्त्यरूपा भवतु सा या त्वां च शशाप ह ।
अधोमर्त्यं सा प्रयातु सन्ति यत्रैव पापिनः ॥ ४० ॥
कलौ तेषां च पापानि ग्रहीष्यति न संशयः ।
O Padmaa! leave this ferocious Saraswati, what will She do to Me. She is foul mouthed impatient ruined constant speaker of offensive language. 36.
She presides over speech and therefore like quarrels always. Just the way Her qualities and powers are, [37th verse], this foul mouthed Saraswati will speak that way only with me.
She want to test Her and my power. [38th verse]. Let all know the prowess between us. Saying this, Ganga cursed Vaani. 39.
O Lakshmi, just like She curses You, let Her also become a river and go to the land of sinner mortal men. There, She will take their sins, in KaliYuga, there is no doubt in this.
इत्येवं वचनं श्रुत्वा तां शशाप सरस्वती ॥ ४१ ॥
त्वमेव यास्यसि महीं पापिपापं लभिष्यसि ।
एतस्मिन्नन्तरे तत्र भगवानाजगाम ह ॥ ४२ ॥
Hearing this speech of Gangaa, Saraswati cursed Her. 41.
You will definitely go to Earth, and take the sins of sinners. Between all this, Bhagawaan came there. 42.
Hari tells Lakshmi, Gangaa and Saraswati, how they shall become rivers on Earth
[Note: here below, Vishnu uses the words "in some parts you become a river" and "in full parts you stay with me" quite often. To understand this read this- "O Lakshmi! in Your part-incarnation(in some parts of Yours) You shall become a river. And in Your full parts (i.e. in whole, completely, in full parts) You shall stay with Me."
I wrote this note, so that reader may not have difficulty in interpreting what is in some parts and in full parts.]
श्रुत्वा रहस्यं तासां च शापस्य कलहस्य च ॥ ४४ ॥
उवाच दुःखितास्ताश्च वाचं सामयिकीं विभुः ।
लक्ष्मि त्वं कलया गच्छ धर्मध्वजगृहं शुभे ॥ ४५ ॥
अयोनिसम्भवा भूमौ तस्य कन्या भविष्यसि ।
तत्रैव दैवदोषेण वृक्षत्वं च लभिष्यसि ॥ ४६ ॥
मदंशस्यासुरस्यैव शङ्खचूडस्य कामिनी ।
भूत्वा पश्चाच्च मत्पत्नी भविष्यसि न संशयः ॥ ४७ ॥
त्रैलोक्यपावनी नाम्ना तुलसीति च भारते ।
कलया च सरिद्भावं शीघ्रं गच्छ वरानने ॥ ४८ ॥
भारतं भारतीशापान्नाम्ना पद्मावती भव ।
Hearing the reason of fight and curses between them,, Lord Vishnu told the three sorrowful Devis, one by one-
O Auspicious Lakshmi! with some of Your parts, go to DharmaDhwaja's house. 45.
You will be born on Earth without any womb, and become his daughter, and there only, due to misfortune, You will become a tree. 46.
Then You will become wife of ShankhaChuuda, who is Indra of Asuras and who has my parts. After this, You will become my wife again, undoubtedly. 47.
You will be known as Tulasi, who purifies the three world, on the land of Bharata. But for now, with some of Your parts, You become a river and quickly go to  BharataBhumi, due to the curse of Bhaarati(Saraswati), and You will be called Padmaavati river there.
गङ्गे यास्यसि पश्चात्त्वमंशेन विश्वपावनी ॥ ४९ ॥
भारतं भारतीशापात्पापदाहाय पापिनाम् ।
भगीरथस्य तपसा तेन नीता सुकल्पिते ॥ ५० ॥
नाम्ना भागीरथी पूता भविष्यसि महीतले ।
मदंशस्य समुद्रस्य जाया जायेर्ममाज्ञया ॥ ५१ ॥
मत्कलांशस्य भूपस्य शन्तनोश्च सुरेश्वरि ।
Then, O Gangaa! You will go in some parts of yours, , to the BharataBhumi, due to curse of Bhaarati(Saraswati), to become the purificator of worlds and to burn the sins of sinners.
O well armed Gangaa! by Bhagiratha's penance You will be brought down, , and will be known as Bhaagirathi, and will be most sanctifying on Earth.
By my orders, You will become wife of Samudra, who is I in parts. .
And also, You will be wife of king Shantanu, who is I manifest in some parts.
गङ्गाशापेन कलया भारतं गच्छ भारति ॥ ५२ ॥
कलहस्य फलं भुङ्क्ष्व सपत्नीभ्यां सहाच्युते ।
स्वयं च ब्रह्मसदने ब्रह्मणः कामिनी भव ॥ ५३ ॥
O Bhaarati(Saraswati)! due to curse of Gangaa, You in some of Your parts go to BharataBhumi, , and O Good Saraswati! bear the fruits of quarrel with these wives. .
And in full parts, You become the wife of Brahma in His abode.
गङ्गा यातु शिवस्थानमत्र पद्मैव तिष्ठतु ।
शान्ता च क्रोधरहिता मद्भक्ता सत्त्वरूपिणी ॥ ५४ ॥
Let Gangaa now go to Shiva's abode and Padmaa(Lakshmi) stay here with Me, as She is calm, never angry, Saatvik and devoted to Me. 45.
So, Saraswati and Ganga fight over Vishnu. Saraswati curses Lakshmi because Lakshmi tries to save Ganga from Saraswati. Ganga curses Saraswati because She cursed Lakshmi without any fault. Then Saraswati in return curses Ganga to be a river. And then Vaasudeva Hari comes to settle down quarrel and directs them how they will incarnate.