It is possible to write any program (i.e. Turing complete) with just one single expression?
I think that your general question is pretty broad, because you'd need to specify what you consider a program (e.g. is the identity function that outputs the input it receives a program?).

It is possible to have some sort of control flow in lambda expressions (there are only expressions in lambda calculus); consider e.g. Church Booleans that enable conditionals.

If you would like to see examples of lambda calculus expressions put to work like regular programs, I can recommend John Tromp's binary lambda calculus. I have been able to evaluate some of his programs using plain β-reductions (examples), so I can say it's definitely possible to write programs even with pure lambda calculus.