Primitive recursive functions can simulate every single step of a Turing machine. In order to prove this, one has to see that a function defined by state table is primitive recursive.
Simply speaking, a partial function $f\colon \mathbb{N} \to \mathbb{N}$ with finite domain is always primitive recursive. That is, if-else and switch-case constructs (in programming languages) can be represented by appropriate primitive recursive functions. How to prove this property?
You can directly write down a definition of the conditional function by primitive recursion:
$$ f(c,x,y) = \begin{cases} y & \text{if }c=0 \\ x & \text{if }c\ge 1 \end{cases} $$
This may not look primitive recursive, but nobody says the recursive case actually has to use the value of $f(c-1,x,y)$.
If you want a switch with more than two cases, just chain a number of if-then-elses together.