The phrase "thanks for your prompt support" is correct and usable in written English. You can use it when expressing your gratitude to someone for their quick and helpful response. For example: "Thank you very much for your prompt support. I really appreciate your help with this issue.".
Thanks for your prompt response.
"And thank you for supporting me as a judge, and thanks for your support when I did the craziest thing of my life and climbed Mount Kilimanjaro".
"For all the other Arabs: thanks for your support," one exhilarated Tunisian tweeted.
Matthew, thanks for your support and I'm glad you were reading.
So again, thanks for your support, and we can't wait to see you next season.
And thanks for your support so far.
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Simone Ivan Conte
Software Engineer at Adobe, UK