Definition of a general rule

rule = Sqrt[a_. * expr_^n_?EvenQ] :> expr^(n/2)*Sqrt[a];

expr = {Sqrt[x^2], Sqrt[x^2 * y], Sqrt[x^2*y^2], Sqrt[x^2*y*z], 
   Sqrt[x^4*y^2 *z]};

Using ReplaceAll (/.)

expr /. rule

{x, x Sqrt[y], x Sqrt[y^2], x Sqrt[y z], x^2 Sqrt[y^2 z]}

However, in general ReplaceRepeated (//.) is needed

expr //. rule

(* {x, x Sqrt[y], x y, x Sqrt[y z], x^2 y Sqrt[z]} *)

Why not use PowerExpand:

PowerExpand[Sqrt[x^2 y]]

x Sqrt[y]

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