Turing completeness is irrelevant, as it only addresses computation. Security has to do with system access, not computational capacity. Brainfuck is Turing complete, but lacks any primitives to do more than read from a single input stream and write to a single output stream. Unless someone hooks those streams up to critical files, you can't use it to attack a system.

Language design actually has a lot of impact on security, because it defines what primitives you have available for interacting with the system. Do you have an arbitrary syscall primitive? Then the language is not going to help you write secure software. Is your only ability to interact with the system via capability objects that must be provided externally to authorize your access? Then you're probably using a language that put a lot of thought into security and will help out quite a lot.