I will be flying from the UK to Kazakhstan via Istanbul's Sabiha Gökçen airport. I'll have an hour layover there, do I need a Turkish transit visa?
I am British and I fly both flights with Pegasus on one ticket.
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In your case you won't be legally entering Turkey at all, because you'll be travelling on one ticket all the way to Kazachstan and you will remain "airside" in the airport terminal.
However, in your case you needn't worry at all, because British citizens have anyway visa-free access to Turkey (https://en.wikipedia.org/wiki/Visa_policy_of_Turkey) for up to 90 days, so even if you have to enter Turkey for whatever reasons, you'll be admitted.