1- What you've been told is right. In your case your limitation period was
2 years and has ended.

2- You can appeal to "Assurence Fund" (="Garanti Fonu") according to
Turkish Trafic Law (Karayolları Kanunu) article 98. Assurence Fund is
responsible of compensating uninsured drivers actions. Though I dont
know if 2 year limitation also applies to Assurence fund applications.