We've been in a similar situation when we had 2 Axolotls and now the same applies for our African giant snail. While we didn't find any specialist in our city, we found some in a larger nearby city.

For my answer, I'll be using turtles and tortoises interchangeably. The answer is not restricted to turtles of course, but applies to any exotic animal.

My ideas (in no particular order):

Search the web for a turtle forum

... and ask there. Maybe add your country's name to the search terms. The chances of finding someone in your vicinity, who can give you recommendations may be higher than on stack exchange (SE).

Suggested search terms (in that order, replace {country} with your country's actual name):

• turtle forum {country}
• reptile forum {country}

Search the web for specialized veterinarians

First try to find one in your city. If you don't find any, check a nearby larger city. If you can't find any there as well, search for your country. You may have to travel for a while to get exotic pets looked at professionally, unfortunately.

You may even find a curated website like this (in German, for Germany).

At least in Germany (maybe in other countries as well), a veterinarian has to take a further training in order to call themselves reptile specialist, since it's usually not covered in the basic vet training. So seeing something like that on a vet's website would be a good sign.

Suggested search terms (in that order, replace {city} with either your city's name or the name of the next larger city):

• turtle veterinarian {city}
• reptile veterinarian {city}
• turtle veterinarian {country}
• reptile veterinarian {country}

Contact veterinarians in / around your city

You can also call or e-mail veterinarians in your city (starting with those you already know) and ask about their skills in regards to reptiles in general or turtles specifically. You can also contact a vet you've found to be specialized in reptiles, if you're not sure if this also applies to turtles or other species (some may only be specialized in geckos, lizards, snakes and such).

From my experience, if asked directly, vets will tell you about their skills in regards to specific animals.

Contact your local / nearest zoo

Zoos cover a wide spectrum of animals and usually know, who to contact about veterinarian problems. Maybe they can give you some hints. Personally, I've not tried this, yet. But I could imagine this would work for turtles.

I'd try to find a customer or info e-mail address (like [email protected]) or a phone number on their website and try to contact that. Sometimes you'll just find a contact form instead. Explain that you got a turtle and can't seem to find a skilled vet for these. Then ask if they can recommend someone.

Maybe (hypothetically) they'll even allow you to bring it in and have them have a look at it. Not sure, though, since you may not be able to bring in your own pets for health and safety reasons.

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When I come up with more ideas, I'll add them here.

Wish you the best of luck.