Proof that if a function is computable with standard programs it is computable with Turing Machines
To write a proof, you need to define your claim precisely. "Standard programs" is not a precise term.

However, if we define "standard program" to be, for example, any valid program in the C programming language, then you can write a proof. To prove this, you describe how to simulate any valid C-program with a Turing machine, or describe a "compiler" that takes a C-program and outputs an equivalent Turing machine. Since the C-language has many features, the proof will be long and tedious, but it is doable. I don't know if anybody has gone through the trouble to write a full formal proof.