Should you carry a lot of cash? That’s up to you. Except for pickpockets, Turkey is a relatively safe country, but nowhere is completely safe. I would say the chance of loss is remote if you carry your stash underneath your clothing in a waist belt, neck pouch, etc. Most hotel rooms of 3 stars or more have small private safes in their guest rooms for valuables
Many people in Turkey will not accept large bills/notes for small payments, so it’s good to juggle your TL cash-on-hand so that you always have some smaller amounts. Get in the habit of paying with bills/notes that are about equal to twice the amount you’re paying (so pay for a TL48 meal with a TL100 note, which gives you TL52 in change). If you give notes that are three times or more than the price, you’re liable to get a dirty look and a request for smaller notes.