I´ll probably get corrected soon, if I´m wrong, and that´s possible, too - but I think none of those countries that you mentioned (Turkey, Bulgaria, Romania, Belarus, Russia and Ukrain) require the carnet as mandatory. I do have personal experience entering Turkey, Bulgaria & Romania without it.
Do note the difference between ALLOWING to use the carnet as a temporary import permit (instead of a country´s own version of that paper - and several countries on that trip may well allow you to do that), or DEMANDING the carnet for entry.
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