It’s actually written 5000 years ago by Lord Sri Vyasadeva as mentioned in itself and other Vedic texts. The Srimad Bhagavatam is the last scripture written by Vyasadeva.
Some evidences which suggests that Bhagavatam is 1800-2000 years old —
The Spiritual guru of Adi Shankaracharya i.e Guadapada wrote the commentary on uttara gita. In that Bhashya or commentary he quoted verses from Srimad Bhagavatam or Bhagavata Purana :
Image ref. and text link—The entire text of Uttara gita can be found in the following link :
Some scholars say that Adi shankaracharya lived in 6th century. Some says in 4th/3rd century. There is no clear date but the dates are between 3rd century and 6th century.
Thus we can clearly understand the dating of Srimad Bhagavatam.
Popularity of Srimad bhagavatam is almost same as Srimad Bhagavad-Gita.
Srimad Bhagavatam has more commetaries than any other vedic scripture except for Bhagavad-gita which has almost same number of commentaries.
There are more than 29+ commentaries written on Srimad Bhagavatam from approx. 10th century :
1) Kardama Kshama (Anvaya)
2) Srila Sridhar Swami’s Bhavartha-Dipika
3) Srila Vamshidhara’s Bhavartha-Dipika-Prakash
4) Srila Jiva Goswami’s Vaishnava Toshani
5) Srila Jiva Goswami’s Brihad-Vaishnava Toshani
6) Sri Sudarshan Suri’s Shukapakshiyam
7) Srila Srinath Chakravarti’s Chaitanya Mata Manjusha
8) Srimad Viraraghavacharya’s Bhagavat-Chandrika
9) Srimad VijaydhvajTirtha’s Pada-Ratnavali
10) Srila Jiva Goswami’s Krama-Sandarbha and Brhat Krama-Sandarbha
11) Srila Vishwanath Chakravarti Thakura’s Sarartha-Darshini 12) Srila Baladeva Vidyabhushana’s Vaishnavanandini
13) Sri Chalarinarayanacharya’s Bhagavat Tatparya Tippani 14) Sri Satyabhinavayati’s Durghata Bhava Dipika
15) Sri Pandhari Narayanacharya’s Durgatoddhara
16) Sri Prabhucharan’s Sri Tippani
17) Sri Purushottam Charan’s Subodhini Prakash
18) Sri Vallabh Maharaja’s Sri Subodhini Lekha
19) Sri Dixit Lalu Bhatta’s Subodhini Yojana
20) Sri Bhagavadeeya Nirbhaya Rama Bhatta’s Subodhini Karika Vyakhya
21) Sri Ganga Sahay’s Anvitartha Prakashika
22) Sri Gopalananda Muni’s Nigudhartha Prakash Vyakhyanam
23) Sri BhagavatPrasadacharya’s Bhakta Manoranjani
24) Sri Harisuri’s Bhakti Rasayanam
25) Sri Shukadev-Acharya’s Siddhanta-Pradipa
26) Srimad Vallabhacharya’s Subodhini
27) Sri Purushottamcharan Goswami’s Subodhini-Prakash
28) Sri GiridharLal Goswami’s Balprabodhini
29)Srila Prabhupada’s Krsna book
There are many others too.
Hare Krishna !