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While visiting Kanchipuram temple in Tamil Nadu, I noticed that they displayed Balrama as 8th avatar of Vishnu, Krishna as 9th and skipped Buddha.
Balrama is actually an avatar of Sheshnaga with a story that Lakshmana wanted to be born older to Rama in the next life.
Why is Buddha not recognised enough in Hinduism like Rama and Krishna? He is the one avatar whose historicity is proven.
Actually, Balarama is an incarnation of Sriman Narayana. The Srimad Bhagavatam designates Balarama as one of the twenty-four incarnations of Sriman Narayana:
In the nineteenth and twentieth incarnations, the Lord advented Himself as Lord Balarama and Lord Krsna in the family of Vrsni [the Yadu dynasty], and by so doing He removed the burden of the world.
At the same time, it is also true that Balarama is an incarnation of Sheshnaga as the Mahabharata claims:
And Baladeva of exceeding strength was a portion of the Naga, Shesha. (Mahabharata 1.61.91)
So the question now arises: How can Balarama be both an incarnation of Sheshnaga and Sriman Narayana?
This question is answered by User @Keshav Srinivasan here.
On the other hand, Buddha is not seen as an incarnation of Vishnu by many because his teachings are considered to be anti-Vedic even though there is no evidence of such.
To add to the other answer which cites Srimad Bhagavatam mentioning that " the Lord advented Himself as Lord Balarama", another source which might probably explain why you saw Balrama as 8th avatar of Vishnu is the Dashavatara Stotram by Jayadeva Goswami, that specifically mentions Lord Balarama as the 8th incarnation of Vishnu.
Here is the corresponding text -
(8)
vahasi vapushi visade vasanam jaladabham
hala-hati-bhiti-milita-yamunabham
kesava dhrita-haladhara-rupa jaya jagadisa hare
0 Kesava! O Lord of the universe! O Lord Hari, who have assumed the form of Balarama, the wielder of the plow! All glories to You! On Your brilliant white body You wear garments the color of a fresh blue rain cloud. These garments are colored like the beautiful dark hue of the River Yamuna, who feels great fear due to the striking of Your plowshare.
