What is the the proper way to define a function from an expression?

Sorry, I was too hasty in comments:

With[{expr = expr}, SetDelayed @@ Hold[f[x_], Expand[expr]]]

SetDelayed @@ Hold is needed because of: Enforcing correct variable bindings and avoiding renamings for conflicting variables in nested scoping constructs

(f[x_] := Expand@#) &@expr

Here are a couple more possibilities. Using Block:

Clear[f]
Block[{Expand}, f[x_] = Expand[expr]];
Definition[f]

f[x_]=Expand[(1+x)^2]

Using With:

Clear[f]
With[{expr = expr, lhs = f[x_]},
    lhs := Expand[expr]
];
Definition[f]

f[x_]:=Expand[(1+x)^2]