DHI and FUE aren't mutually exclusive.
DHI is the implantation method where grafts are directly placed into the scalp with a pen (it basically creates the incisions and implants the graft at the same time). It's gotten a lot of praise recently but admittedly a lot of it is just marketing.
FUE is the extraction method where grafts are individual extracted with a punch. The other extraction method is FUT where a strip is taken from the back of the scalp.
Im not really sure why they would say it wouldnt suit you if that is their standard operating technique. If that is your pic it looks like youre a higher NW. Id suggest you do some consults with those clinics that specialise in mega sessions; H&W, Hattingen and Eugenix.
IMO there aren't many reliable places to go in Turkey, HLC, Keser and Pekiner are the stand outs, with Demisroy the most reliable for those on a budget. Id suggest you check out more patient reviews as this industry can be a bit tough to navigate (Turkey has the highest number of black market clinics in the world).