Is it correct to say "I want to continue/further my master's in the UK"?
- I want to continue my master's in the UK.
- I want to further my master's in the UK.

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'Continue my masters' but this means you will start it in Thailand?
 or 'further my studies/education'
:)

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I want to continue my master's.. means you have started it already somewhere else, you want to transfer to a British university to study for it there instead.

I want to further.. makes no sense.

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Yes, these are correct. They essentially mean the same thing, so it is okay to use both.

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