Did Lord Krishna come on earth via a normal birth or his birth was a supernatural event ?
What is the exact story related to it?
According to Vishnu Purana, Book 4, Chapter 15, Lord Krishna birth was normal. He has taken birth from the womb of Devaki.
Next, the divine Vishńu himself, the root of the vast universal tree, inscrutable by the understandings of all gods, demons, sages, and men, past, present, or to come, adored by Brahmá and all the deities, he who is without beginning, middle, or end, being moved to relieve the earth of her load, descended into the womb of Devakí, and was born as her son Vásudeva.
According to Shrimada Bhagavatam also Lord Krishna birth was normal. Shrimada Bhagavatam, Canto 10, Chapter 2, Devaki kept him in herself.
SB 10.2.18 — Thereafter, accompanied by plenary expansions, the fully opulent Supreme Personality of Godhead, who is all-auspicious for the entire universe, was transferred from the mind of Vasudeva to the mind of Devakī. Devakī, having thus been initiated by Vasudeva, became beautiful by carrying Lord Kṛṣṇa, the original consciousness for everyone, the cause of all causes, within the core of her heart, just as the east becomes beautiful by carrying the rising moon.
SB 10.2.19 — Devakī then kept within herself the Supreme Personality of Godhead, the cause of all causes, the foundation of the entire cosmos, but because she was under arrest in the house of Kaṁsa, she was like the flames of a fire covered by the walls of a pot, or like a person who has knowledge but cannot distribute it to the world for the benefit of human society.
And then, in Shrimada Bhagavatam, Canto 10, Chapter 3, it is explicitly mentioned that Lord Krishna has taken birth from womb.
SB 10.3.31 — At the time of devastation, the entire cosmos, containing all created moving and nonmoving entities, enters Your transcendental body and is held there without difficulty. But now this transcendental form has taken birth from my womb. People will not be able to believe this, and I shall become an object of ridicule.
Harivamsa Purana, Vishnu Parva, Chapter 4 also confirm that Lord Krishna birth was normal.
After her (seventh) conception was aborted, devakI conceived (for the eighth time), for which, kaMsa had destroyed her seven conceptions. 2.4.8
The ministers (of kaMsa) guarded that conception with careful efforts. But then, hari (viShNu) was residing in her womb at his self will. 2.4.9
The same day, yashodA, conceived nidrA who carries out the instructions of viShNu and who manifested from viShNu's body. 2.4.10
Before the completion of the normal period of pregnancy, the women, devakI and yashodA delivered in the eighth month at the same time. 2.4.11
A girl was born to yashodA at the same time of the same night when kR^iShNa, who elevates the (fame of) vR^iShNi race was born. 2.4.12
There was a king called Kansa, the ruler of Mathura. He was a very evil ruler and treated his subjects poorly.He had a sister called Devaki. She married the Yadava, Vasudeva. Kansa was okay with the marriage. But when they were returning to Mathura, they heard this prophecy from the skies, that said that the 8th son of Vasudeva and Devaki will kill Kansa.
Kansa was furious and imprisoned both Devaki and Vasudeva. So one by one Kansa slew all 6 kids of Devaki, who were all born in the prison. The 7th kid was transferred to the womb of Vasudeva's other wife Rohini. That child is Balarama. Then was the time for the 8th kid. Before the baby's birth, Lord Vishnu came to Vasudeva and told that he is going take birth as their child and to take the child and exchange it with the child born to Nanda, a friend of Vasudeva. So that night when the baby was born, Vasudeva's cuffs released themselves and when he had to go through the waters, they parted and finally somehow he exchanged the child. When Kansa came to destroy the child he ridiculed it for being a girl and threw the child high up in the air when the child transformed to Goddess Kali and told him that 'The one who is going to kill you has already taken birth'. That's how Lord Krishna was born.
Source: Birth of Lord Krishna
According to Bhagavatam it was not a normal biological birth.
9-10. Vasudeva saw that wonderful child of lotus-like eyes, endowed with four arms, bearing a conch and wielding aloft a mace and a discus (and a lotus); having (on his chest) the Srivatsa mark (of curl of golden hair), with the Kaustubha gem (hanging from his neck but) beautified by his neck; clad in yellow silken garments, possessing the charming complexion of rain-bearing cloud ; his profuse locks of hair bathed in the lustre of his crown and ear-rings studded with invaluable vaiddrya-gems (lapis lazuli); resplendent with his exquisite and valuable zone, armlets, bracelets and other ornaments.
(Having first embraced the Lord as his son)
Vasudeva said :
How blessed I am that I have seen you directly — you the Supreme Person, transcendental to and distinct from Prakfti , absolute wisdom and supreme bliss incarnate, and (as inner controller) the witness of all minds and intellects.
(As a matter of fact you never entered the womb of Devaki for) you are factually of such absolute nature
The Vishnu Purana says:
When Devakí was pregnant the seventh time, Yoganidrá (the sleep of devotion), sent by Vishńu, extricated the embryo from its maternal womb at midnight, and transferred it to that of Rohińí; and from having been thus taken away, the child (who was Balaráma) received the name of Sankarshańa. Next, the divine Vishńu himself, the root of the vast universal tree, inscrutable by the understandings of all gods, demons, sages, and men, past, present, or to come, adored by Brahmá and all the deities, he who is without beginning, middle, or end, being moved to relieve the earth of her load, descended into the womb of Devakí, and was born as her son Vásudeva. Yoganidrá, proud to execute his orders, removed the embryo to Yasodá, the wife of Nanda the cowherd.
There are two embryo-transfers. One has to understand that the 7th pregnancy was removed to make room for the eighth without the normal process of conception.