How to bake lokum rolls well?

I would start by rolling the pastry dough thinner; I often have my dough too thick the first time or two that I work with a recipe. Keep the dough cold to make it easier to work with.

Next, I would try different oven times and temperatures. Try a temperature 25-50 degrees (F) higher, and pull the rolls out 30 seconds to one minute sooner. The nice thing about playing with oven times and temps is that you can go through many tests quickly by just putting a few of the items in for each time/temp combination. (Extra mass in the oven, such as a pizza stone or even some bricks, will help keep the oven temp more steady.)

If the outside of your rolls are getting over done before the inside is done, try chilling the rolls in the refrigerator for 30 minutes or so before baking.

There are two techniques I can think of, that might help. You can try one or the other, or both, as it suits your fancy.

First, you mention that a thinner roll of pastry will affect the ratio of filling? Make the whole log thinner. Take that same square of dough, make it into a longer rectangle. take the same cylinder of turkish delight, make it into a longer cylinder to match. Each pastry has the same amount of each, so the ratio will be the same - but the pastry will bake more quickly and thoroughly. You will have to shorten cooking times, and the thinner rolls will also dry faster if you're not careful, but it can be done.

The second technique is a little more complicated. It helps to have a pizza stone or baking tile, and to make sure the oven is well preheated. Bake them at a somewhat higher temperature, with a shorter baking time, as KatieK suggests - but when the rolls are still underdone, turn off the oven. The residual heat will continue baking the rolls, but since the oven coils are off, the heat won't be directional to brown the upper part of the rolls more quickly than the inside.

The indirect heat means they will take longer during this phase (for example, if you turned off the oven with 5min to go, it may take 10 min or more to actually finish), and it may take some experimentation to find the correct results - if the rolls aren't close enough to being done, the oven may cool before they finish cooking, and if they're too close, they might overcook anyway. But it does give finer control and a bit more flexibility over the top being browned versus a dish being cooked through.

Again, there's a risk the pastry will dry a little more quickly if you're not careful. Depending on the recipe, I might be very careful about cooking times to avoid the effect, keep a dish of water in the oven so the environment is moister, or simply brush the outside in water at some point. If you brush the ops in water when you turn the oven off, it will also help retard any browning or overcooking of the tops of the rolls.