A dip in Ganga river does remove all the sins when it is done with śraddhā and bhakti. To do so, the words śraddhā and bhakti should be understood.
Bhakti is well known to everyone. Narada Bhakti Sūtras talks about nine forms of Bhakti. Ādi Śaṅkarācārya says mokṣa sādhana sāmagryāṃ bhaktireva garīyasi meaning Bhakti is the main tool among all in the practice of Moksha.
Ādi Śaṅkarācārya defines śraddha as
śāstrasya guruvākyasya satyabuddhyāvadhāraṇā । sā śraddhā kathitā
sadbhiḥ yayā vastūpalabhyate ॥
The firm belief that Śāstra and Guru vākya are true is called śraddhā
which is bestower of anything.
Śāstra is nothing but vedas, purāṇas, smritis etc.
The Puranas do not contradict anyway. In the Chapter Six called Tīrthādhyāya from Kashi Khanda Purvardha of Skanda Puranam, Agastya Mahāmuni says to Lopāmudra as
श्रुणुतीर्थानिगदतोमानसानिममानघे । येषुसम्यङ्नरः
स्नात्वाप्रयातिपरमांगतिम् ॥ २९ ॥
O sinless lady, listen even as I enumerate the mental holy spots. By
scrupulously taking the holy dips in them a man attains the greatest
सत्यं तीर्थं क्षमातीर्थं तीर्थमिन्द्रियनिग्रहः । सर्वभूतदयातीर्थं
तीर्थमार्जवमेवच ॥ ३० ॥
दानंतीर्थं दमस्तीर्थं सन्तोषस्तीर्थमुच्यते । ब्रह्मचर्यं परंतीर्थं
तीर्थंचप्रियवादिता ॥ ३१ ॥
ज्ञानंतीर्थंधृतिस्तीर्थं तपस्तीर्थमुदाहृतम् । तीर्थानामपि तत्तीर्थं
विशुद्धिर्मनसः परा ॥ ३२ ॥
Truth, forbearance, control of organs of sense and action, compassion
for all beings, straightforwardness, dānam, self-restraint,
contentedness, celibacy, speaking pleasant words, knowledge, courage,
penance are all tīrthās. The absolute purity of mind is the holiest of
all holy tīrthās.
न जलाप्लुतदेहस्यस्नानमित्यभिधीयते । सस्नातोयोदमस्नातःशुचिःशुद्धमनोमलम्
॥ ३३ ॥
Holy ablution is not mere drenching of the body in water. He who has
the purificatory bath of mental control, is really the one who has had
his holy ablution; he is really clean; he has removed all the
impurities of the mind.
यो लुब्धः पिशुनः क्रूरोदाम्भिकोविषयात्मकः । सर्वतीर्थेष्वपि स्नातः
पापो मलिन एवसः ॥ ३४ ॥
One who is covetous, slanderous, cruel, a hypocrite and sensualist is
sinful and dirty, even if he takes a plunge in all the sacred waters.
न शरीरमलत्यागान्नरो भवतिनिर्मलः । मानसे तु मले त्यक्तेभवत्यन्तः
सुनिर्मला ॥ ३५ ॥
A man does not become free from impurities only by doing away with the
bodily dirt. If mental dirt is wiped away, one becomes internally
जायन्ते च म्रियन्तेचजलेष्वेजलौकसः । न च गच्छन्तितेस्वर्गमविशुद्धमनोमला
॥ ३६ ॥
Aquatic beings are born and die in water. But they do not go to heaven
because their mental impurities have not been cleansed.
चिगयेष्वतिसंरागो मानसोमल उच्यते । तेष्वेव हि विरागोस्यनैर्मल्यं
समुदाहृतम् ॥ ३७ ॥
Too much of attachment to sensual objects is called mental impurity.
Non-attachment to them is cited as freedom from impurities.
चित्तमन्तर्गतं दुष्टंतीर्थस्नानान्न शुद्ध्यति ।
शतशोथजलैर्धौतंसुराभाण्डमिवाशुचि ॥ ३८ ॥
If the mind within is defiled, it does not become purified by taking a
holy dip in sacred waters. It is impure like a pot of toddy washed
hundreds of times in water.
दानमिज्यातपः शौचंतीर्थसेवाश्रुतं तथा ।
सर्वाण्येतानितीर्थानियदिभावोननिर्मला ॥ ३९ ॥
If the mental predilection is not pure, all these, viz. liberal gifts,
sacrifice, penance, cleanliness, piligrimage and learning, becomes
निगृहीतेन्द्रियग्रामोयत्रैवचवसेन्नरः । तत्रतस्य कुरुक्षेत्रं
नैमिषंपुष्कराणिच ॥ ४० ॥
Wherever a man who has curbed all his organs of sense and action
lives, there he has his Kurukshetra, Naimisha, Pushkara etc.
ध्यानपूते ज्ञानजले रागद्वेषमलापहे । यः स्नाति मानसे तीर्थे स
यातिपरमांगतिम् ॥ ४१ ॥
He who takes his holy bath in the mental holy spot that is cleansed
with perfect knowledge, that has meditation for its waters and that
removes the dirt of attachment and hatred, attains the greatest goal.
एतत्ते कथितं देवि मानसंतीर्थलक्षणम् । भौमानामपि
तीर्थानांपुण्यत्वेकारणंश्रुणु ॥ ४२ ॥
These characteristics of the mental tīrthās have been recounted to
you, O noble lady. Listen to the cause that brings about sanctity to
worldly holy spots.
यथा शरीरस्योद्देशाः केचिन्मेध्यतमाः स्मृताः । तथापृथिव्यामुद्देशाः
केचित्पुण्यतमाः स्मृताः ॥ ४३ ॥
Just as there are certain excellent parts of the body considered very
holy and pure, so also some regions of the earth are considered most
प्रभावादद्भुताद्भमेः सलिलस्य च तेजसा ।
परिग्रहान्मुनीनाञ्चतीर्थानांपुण्यतास्मृता ॥ ४४ ॥
The meritoriousness of tīrthās is considered to be a result of the
mysterious influence of the ground, of the water and of the fire
thereof as well as of the backing and acceptance by sages.
तस्माद्भौमेषु तीर्थेषुमानसेषुचनित्यशः । उभयेष्वपियः स्नातिसयाति
परमांगतिम् ॥ ४५ ॥
Hence he who takes his holy bath in both the holy spots i.e., the
worldly holy spots and the mental holy spots attains the greatest
Also Jagadguru Shankaracharya Abhinava Vidyateertha Mahaswami of Sringeri Sarada Peetham has often spoken of the efficacy of bathing in Gaṅgā as a means of purification (Source: Exalting Elucidations, Page 52 freely downloadable from here). He has said,
“Gaṅgā does purify people. When the faith of the person who resorts to
her is intense, he is greatly benefitted by her. This, however, does
not mean that one will be redeemed if one commits sins deliberately
and then bathes in the Gaṅgā.”
In another conversation with Mahaswami he also told as (D: Devotee, A: Acharya)
D: It is believed that the river Gaṅgā effaces one’s sins. Does it
actually do so?
A: Yes. Why is there any doubt about it?
D: If Gaṅgā removes one’s sins, can one commit any sin and then get
rid of it by bathing in the Gaṅgā?
A: A dip in the Gaṅgā will remove those sins that are unknowingly done
or knowingly done once for which we are repentant. It is not fair to
expect Gaṅgā to free one from an intentional sin done with the
presumption, “I will bathe in the Gaṅgā and get rid of it.” However,
just because of the above statement, one need not have even an iota of
doubt regarding the greatness of Gaṅgā. Crores of people travel long
distances with the firm faith that they will be purified by a bath in
the Gaṅgā. Īśvara does not let their faith go unrewarded.
There is a story in this connection. Once, Pārvatī asked Śiva whether
Gaṅgā would remove the sins of all. As an answer, Śiva took on the
appearance of an old man and took Pārvatī, in the garb of His aged
wife, to the banks of Ganges. He told Her that He would pretend to be
drowning and that She should raise an alarm. Accordingly, She screamed
to attract the attention of the passers-by and sought the rescue of
Her husband. The people around rushed to His rescue. The disguised
Pārvatī, however, stopped them and, as earlier instructed by Śiva,
said, “Only that person who has not the slightest trace of sin can
touch My husband.” Only one among the throng dived in without the
least hesitation and rescued Him. Surprised, Pārvatī asked the rescuer
if he was free from all sins. The man replied, “Did I not have a dip
in the Gaṅgā before saving your husband? My sins vanished at that very
instant.” Śiva then told Pārvatī, “What doubt need be there that sins
are washed away if one has faith like this person?”
So a dip in Ganges along with mental purity (faith and devotion) gives the real benefit of greater good.
Hope this helps you.