Under no circumstances should you leave the country as you are presently in immigration removal proceedings. If you leave the country while in proceedings you will be considered to have "self-deported" and will not be allowed to return.
While Puerto Rico is the United States and travel there should not cause an issue - travel to Turkey is absolutely out. And if you were my client, I would advise you to just stay at home until your case is resolved.
Despite Puerto Rico being part of the United States, it is not worth the risk to fly as CBP (even with interstate travel) has been making mistakes and reporting people to immigration at passport check-in (security). Immigration is very quick to take people into custody first, ask questions later. It is not worth the possible pitfalls to take the trip.
My advice is that you should not leave the US. You can travel to different US states, but not out of the country.
How did you file for asylum? If you file for asylum while you are in the US, which is affirmative asylum, then you can generally stay in the US while your case is pending. But if you first arrive into a US port of entry and the officers determine you have an asylum claim, then you will be detained until the asylum proceedings are over (defensive asylum).
If you travel out of the US and return, then an officer may detain you until the asylum process is over, like in a defensive asylum. I have not run into a case like this but I imagine that this would be possible. So, my advice is that you should not risk leaving the US.
What is most critical is that you do NOT travel to your country that is the basis for your asylum claim! If there are changed conditions in the country that enable you to return there without a well founded fear of persecution (harm), then your asylum claim will fail.
It is essential for you to see your judge for the individual hearing on the scheduled July date. Do not miss it! With a lawyer, your have a much better chance to prevail on your asylum claim!
In the future, if you get asylum granted, lawyers differ as to whether you can go back to your country that was the basis of the asylum claim. Here is the memo describing the policy from the government, which can be interpreted in different ways: http://www.uscis.gov/files/pressrelease/AsylumTravel122706FS.pdf.
Again, the safest option would be to avoid going back to your country that was the basis of the asylum.