What is wrong with this sentence: "This is an obstacle to start in Turkey"

The problem with the sentence is nothing syntactical. It is, rather, fatally unclear. We don't know what the demonstrative pronoun "this" refers to, so we're in the dark about the subject of the sentence, which appears to be part of a conversation.

"To start in Turkey" is equally murky. Starting what? Is the nature of the obstacle physical, political, economic, or something else? "I'd start in general poultry if I were you. To start in turkey is an obstacle--too small a market, too small a margin."

In short, we have no idea what the sentence means and no way of finding out with the information given.

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)

For example:

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.
• You 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.

An obstacle requires to with a gerund, not an infinitive. There is no reason for this: it just happens to be a fact about English at the moment. (It might have changed fifty years from now).

As another answer says, if you say an obstacle to start .., the to start cannot be read as a complement of obstacle, and must be interpreted in a different way.

an obstacle to starting in Turkey

is fine as a noun phrase.