Let $2^x=t$

$5+\left|t-1\right|=t^2-2t$

$\Rightarrow \underset{g\left(t\right)}{\left|t-1\right|}=\underset{f\left(t\right)}{\left(t^2-2t-5\right)}$

Since, $2^x>0 \forall x\in R\Rightarrow t>0$

From the graph, we have

It is clear from the graph that the two graphs have only one point of intersection.

So, number of real roots of the given equation is $1.$