If you say
This is an obstacle to start in Turkey.
"to start" looks like an infinitive of purpose, but this is contrary to the meaning of the sentence. "This" prevents something from happening, it does not make it happen. Obstacle to starting is correct, because "to" is a preposition (it is not part of a to-infinitive). Prepositions are followed by nouns, and in this case the gerund "starting" functions as a noun. We are asking the question: "This is an obstacle to what?", not "What kind of obstacle is this?" and even less "What purpose does this obstacle have?" The answer of "To what?" is "To starting in Turkey", where "starting" can be replaced by nouns like commencement or debut (these are just examples, I am not saying they are perfect synonyms):
This is an obstacle to starting in Turkey. This is an obstacle to our debut in Turkey.
"an obstacle" can be replaced by a verb like "prevent":
This prevents our debut in Turkey. This prevents us from starting in Turkey.
As for "obligation to do something" that was discussed in the comments, this is an idiomatic phrase:
have a duty/responsibility/obligation etc (to do something) which is used for saying you must do something (Macmillan)
Buyers have no legal obligation to disclose personal financial information.
Here you would ask the question, "Which obligation?"
Cambridge dictionary shows that "obligation" is followed by a to infinitive:
[ + to infinitive ]
- If you have not signed a contract, you are under
no obligation to (= it is not necessary to) pay them any money.
have a legal obligation to (= the law says you must) ensure your child
receives an education.
"Obligation" can be followed by the preposition TO, but TO needs to be followed by a noun (phrase):
One has an obligation to one's friends. (To whom? To one's friends.)
Having said all that, I also checked Gngram 1 and Gngram 2, and I was surprised to see that it found instances of "obstacle to do" and "obligation to doing", even if they are very rare. However, if you use obstacle + to + Gerund and obligation + to infinitive, you will always be correct.