I received a letter from the IRB on Jan which states that my claim will be processed under the expedited system. My question is, how long does it take, for those who have been through the same situation, to hear back from them?
Expedited process doesn't necessarily mean a faster process. It means your case will be judged without a hearing within or beyond the average processing time which is around 1-2 years currently.
On what grounds are you claiming asylum? If you claimed asylum so you have to pay lower tuition fee, you should be prepared for a quick removal from Canada. You had an approved study permit, you could have chosen to study, get a PGWP and then eventually obtain PR like everyone else but the path you chose is most likely going to end you back up in Turkey with a ban from Canada and maybe even United States.