The British pronounce "thirsty" as "thur-sti", with a subtle 'r' sound and a short 'i' sound.
The YouTube video reference demonstrates the British pronunciation of the word "thirsty." In British English, the 'r' sound after the 'th' and before the 's' is often present but quite subtle, compared to some other accents. The vowel sound in the second syllable "-sti" is a short "i" as in "sit". Therefore, a common British pronunciation of "thirsty" sounds like "thur-sti".