The ranking of Puranas

Suta Goswami spoke all eighteen major Puranas at Naimisharanya, and the sages present accepted them as authentic. Nonetheless, three groups of six Puranas each are meant for three different audiences, depending on which of the three modes of nature predominates each audience. But for each individual Purana the situation is more complex because most Puranas display some mixture of the modes. For example, the pastimes of Lord Krishna and those of Lord Ramachandra, which are in the mode of pure goodness (suddha satva), are described to some extent in most of the Puranas.

Despite vedic references, sometimes people criticize ISKCON saying that ISKCON ignores other Gods and other Puranic literature beyond Srimad Bhagavatam. I think ISKCON maintains vedic integrity (thus authentic in my opinion) than any so-called new-age advaita vadis.

The way to reconcile all the contradictory nature of the scriptures is by following in the footsteps of Mahajans (Mahajano yena gata sapamta). This is the criterion to understand Godhead or anything spiritual. On our own, reading scriptures is not permitted in the Vedas. It should be studied under a bonafide spiritual master following the parampara system.

Lord Shiva is one of the Mahajans and we follow His instructions. Below are His words to His wife Mother Parvati. It is taken from Padma Purana Uttara Khanda 236.18 - 20.

vaishnavam naradiyam cha
tatha bhagavatam shubham
garudam cha tatha padmam
varaham shubha-darshane
sattvikani puranani
vijneyani shubhani vai

"O beautiful one, the Vishnu Purana, the Narada Purana, the auspicious Bhagavata Purana, and the Garuda, Padma, and Varaha Puranas all belong to the mode of goodness. They are all considered auspicious.

brahmandam brahma-vaivartam
markandeyam tathaiva cha
bhavishyam vamanam brahmam
rajasani nibodhata

"Know that the Brahmanda, Brahma-vaivarta, Markandeya, Bhavishya, Vamana, and Brahma Puranas belong to the mode of passion.

matsyam kaurmam tatha laingam
shaivam skandam tathaiva cha
agneyam cha shad etani
tamasani nibodhata

"And know that these six Puranas belong to the mode of ignorance: the Matsya, Kurma, Linga, Shiva, Skanda, and Agni Puranas.

Reading these puranas come with its consequences. The whole point is to elevate one to higher consciousness and not degrade. However, one who is against God will degrade. Below Lord Shiva mentions the result of reading such scriptures.

satvika moksadah prokta 
rajasah sarvadashubhah 
tathaiva tamasa devi niraya prapti hetavah
tathaiva smrtayah prokta trsibhis trigunanvitah

Sattvika Puranas are said to give liberation, rajasika Puranas is everything inauspicious and tamasika Puranas cause one to go to hell, oh Devi!. Thus they should be remembered as endowed by three gunas. - Padma Purana Uttara Khanda 236.21-22

In these puranas stories are narrated in such a way to incite devotion to a particular god. Satvik puranas invoke devotion to Vishnu, rajasic to Brahma and tamasic to Shiva. As a result, one will find contradictions in Supremacy. Below is a criterion given by Lord Shiva to reconcile such contradiction.

ata uktam skande shanmukham prati sri-sivena
siva-sastresu tad grahyam
bhagavac-chastra-yogi yat
paramo visnur evaikas
taj jnanam moksa-sadhanam
sastranam nirnayas tv
esas tad anyan mohanaya hi. iti.

The statements of the Siva scriptures should be accepted only when they agree with the Visnu scriptures. Lord Visnu is the only Supreme Lord, and knowledge of Him is the path to liberation. That is the conclusion of all the scriptures. Any other conclusions are meant only to bewilder the people."  - Lord Siva to Karttikeya - Skanda Purana

In conclusion, if one is serious to ascertain the identity of the Supreme Personality of Godhead without doubt, then one must approach a bonadide tattva darshi (not self proclaimed so-called sadhus).

Vaisampayana rishi who is a great sage has this to say;

ankhyam yogas ca sanatane dve
vedas ca sarve nikhile 'pi rajan
sarvaih samastair risibhir nirukto
narayanam visvam idam puranam. iti.

Sankhya and yoga are both eternal. All the Vedas are also eternal. All the sages declare that Lord Narayana is the ancient Supreme Personality of Godhead, the master of all the worlds - Moksa-dharma, Narayaniyopakhyana

Hare Krishna