I heard a lot that modern computers are based on Turing machine. I just cannot build a bridge from a conceptual Turing Machine to a real modern computer. Could someone help me build this bridge?

Below is my current understanding.

I think the computer is a big general-purpose Turing machine. Each program we write is a small specific-purpose Turing machine. The classical Turing machine do its job based on the input and its current state inside and so do our programs.

Let's take a running program (a process) as an example. We know that in the process's address space, there's areas for stack, heap, and code. A classical Turing machine doesn't have the ability to remember many things, so we borrow the concept of stack from the push-down automaton. The heap and stack areas contains the state of our specific-purpose Turing machine (our program). The code area represents the logic of this small Turing machine. And various I/O devices supply input to this Turing machine.