Is conditional branching a requirement of Turing-completeness?

Question

I\'ve been searching the web and I\'m finding somewhat contradictory answers. Some sources assert that a language/machine/what-have-you is Turing complete if and only if it has

Answer 1

If a machine can branch, then yes it's considered Turing complete.

The reason is having conditional-branching automatically makes any computer Turing complete. However, there are also machines that can't jump branch or even IF but are still considered Turing complete.

Processing is just the process of identifying inputs in-order to select outputs.

Branching is one way to mentalize this process, the condition of the jump is what can classify inputs, the place you branch to stores the correct output for that input.

So finally, to clarify things:

If you have conditional branching your computer is necessarily computationally equivalent to a Turing machine. However, there are plenty of other ways for a computer to achieve Turing completeness (lambda, IF's, CL).