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NOTE: the answer below refers to a somewhat different version of the OP's question.

Refine seems to be your friend here:

bop = Sqrt[x - 2]; Refine[Conjugate@bop, x < 2] (* Out: -I Sqrt[2 - x] *) 

Notice that, by implicating $x$ in an inequality, the system automatically includes the assumption that it must therefore be real, so only the inequality condition is necessary.

This is one more case that highlights that the user's and the system's concepts of "simpler" may not coincide (see also the discussion in my answer to Advice for Mathematica as Mathematician's Aid).

Simplify and its friends try to minimize the complexity of an expression, as measured mostly (but not uniquely) by the LeafCount of the expression. It so happens that in your case the LeafCount of your desired form is higher than that of the original Conjugate expression:

LeafCount[-I Sqrt[2 - x]] (* Out: 13 *) LeafCount[Conjugate[Sqrt[-2 + x]]] (* Out: 8 *) 

You can make sense of this somewhat unexpected result by looking at the FullForm of these expressions:

FullForm[-I Sqrt[2 - x]] (* Out: Times[Complex[0, -1], Power[Plus[2, Times[-1, x]], Rational[1, 2]]] *) FullForm[Conjugate[Sqrt[-2 + x]]] (* Out: Conjugate[Power[Plus[-2, x], Rational[1, 2]]] *) 

So, even though Refine is explicitly attempted by Simplify (as stated in the details of the documentation to Refine), its result is discarded in this case because the system deems it "more complex", and Simplify returns the unmodified input as the best form it can come up with!

NOTE: the answer below refers to a somewhat different version of the OP's question.

Refine seems to be your friend here:

bop = Sqrt[x - 2]; Refine[Conjugate@bop, x < 2] (* Out: -I Sqrt[2 - x] *) 

Notice that, by implicating $x$ in an inequality, the system automatically includes the assumption that it must therefore be real, so only the inequality condition is necessary.

This is one more case that highlights that the user's and the system's concepts of "simpler" may not coincide (see also the discussion in my answer to Advice for Mathematica as Mathematician's Aid).

Simplify and its friends try to minimize the complexity of an expression, as measured mostly (but not uniquely) by the LeafCount of the expression. It so happens that in your case the LeafCount of your desired form is higher than that of the original Conjugate expression:

LeafCount[-I Sqrt[2 - x]] (* Out: 13 *) LeafCount[Conjugate[Sqrt[-2 + x]]] (* Out: 8 *) 

You can make sense of this somewhat unexpected result by looking at the FullForm of these expressions:

FullForm[-I Sqrt[2 - x]] (* Out: Times[Complex[0, -1], Power[Plus[2, Times[-1, x]], Rational[1, 2]]] *) FullForm[Conjugate[Sqrt[-2 + x]]] (* Out: Conjugate[Power[Plus[-2, x], Rational[1, 2]]] *) 

So, even though Refine is explicitly attempted by Simplify (as stated in the details of the documentation to Refine), its result is discarded in this case because the system deems it "more complex", and Simplify returns the unmodified input as the best form it can come up with!

Refine seems to be your friend here:

bop = Sqrt[x - 2]; Refine[Conjugate@bop, x < 2] (* Out: -I Sqrt[2 - x] *) 

Notice that, by implicating $x$ in an inequality, the system automatically includes the assumption that it must therefore be real, so only the inequality condition is necessary.

This is one more case that highlights that the user's and the system's concepts of "simpler" may not coincide (see also the discussion in my answer to Advice for Mathematica as Mathematician's Aid).

Simplify and its friends try to minimize the complexity of an expression, as measured mostly (but not uniquely) by the LeafCount of the expression. It so happens that in your case the LeafCount of your desired form is higher than that of the original Conjugate expression:

LeafCount[-I Sqrt[2 - x]] (* Out: 13 *) LeafCount[Conjugate[Sqrt[-2 + x]]] (* Out: 8 *) 

You can make sense of this somewhat unexpected result by looking at the FullForm of these expressions:

FullForm[-I Sqrt[2 - x]] (* Out: Times[Complex[0, -1], Power[Plus[2, Times[-1, x]], Rational[1, 2]]] *) FullForm[Conjugate[Sqrt[-2 + x]]] (* Out: Conjugate[Power[Plus[-2, x], Rational[1, 2]]] *) 

So, even though Refine is explicitly attempted by Simplify (as stated in the details of the documentation to Refine), its result is discarded in this case because the system deems it "more complex", and Simplify returns the unmodified input as the best form it can come up with!

NOTE: the answer below refers to a somewhat different version of the OP's question.

Refine seems to be your friend here:

bop = Sqrt[x - 2]; Refine[Conjugate@bop, x < 2] (* Out: -I Sqrt[2 - x] *) 

Notice that, by implicating $x$ in an inequality, the system automatically includes the assumption that it must therefore be real, so only the inequality condition is necessary.

This is one more case that highlights that the user's and the system's concepts of "simpler" may not coincide (see also the discussion in my answer to Advice for Mathematica as Mathematician's Aid).

Simplify and its friends try to minimize the complexity of an expression, as measured mostly (but not uniquely) by the LeafCount of the expression. It so happens that in your case the LeafCount of your desired form is higher than that of the original Conjugate expression:

LeafCount[-I Sqrt[2 - x]] (* Out: 13 *) LeafCount[Conjugate[Sqrt[-2 + x]]] (* Out: 8 *) 

You can make sense of this somewhat unexpected result by looking at the FullForm of these expressions:

FullForm[-I Sqrt[2 - x]] (* Out: Times[Complex[0, -1], Power[Plus[2, Times[-1, x]], Rational[1, 2]]] *) FullForm[Conjugate[Sqrt[-2 + x]]] (* Out: Conjugate[Power[Plus[-2, x], Rational[1, 2]]] *) 

So, even though Refine is explicitly attempted by Simplify (as stated in the details of the documentation to Refine), its result is discarded in this case because the system deems it "more complex", and Simplify returns the unmodified input as the best form it can come up with!

Refine seems to be your friend here:

bop = Sqrt[x - 2]; Refine[Conjugate@bop, x < 2] (* Out: -I Sqrt[2 - x] *) 

Notice that, by implicating $x$ in an inequality, the system automatically includes the assumption that it must therefore be real, so only the inequality condition is necessary.

Refine seems to be your friend here:

bop = Sqrt[x - 2]; Refine[Conjugate@bop, x < 2] (* Out: -I Sqrt[2 - x] *) 

Notice that, by implicating $x$ in an inequality, the system automatically includes the assumption that it must therefore be real, so only the inequality condition is necessary.

This is one more case that highlights that the user's and the system's concepts of "simpler" may not coincide (see also the discussion in my answer to Advice for Mathematica as Mathematician's Aid).

Simplify and its friends try to minimize the complexity of an expression, as measured mostly (but not uniquely) by the LeafCount of the expression. It so happens that in your case the LeafCount of your desired form is higher than that of the original Conjugate expression:

LeafCount[-I Sqrt[2 - x]] (* Out: 13 *) LeafCount[Conjugate[Sqrt[-2 + x]]] (* Out: 8 *) 

You can make sense of this somewhat unexpected result by looking at the FullForm of these expressions:

FullForm[-I Sqrt[2 - x]] (* Out: Times[Complex[0, -1], Power[Plus[2, Times[-1, x]], Rational[1, 2]]] *) FullForm[Conjugate[Sqrt[-2 + x]]] (* Out: Conjugate[Power[Plus[-2, x], Rational[1, 2]]] *) 

So, even though Refine is explicitly attempted by Simplify (as stated in the details of the documentation to Refine), its result is discarded in this case because the system deems it "more complex", and Simplify returns the unmodified input as the best form it can come up with!