To expand on an earlier answer, I will plagiarize an answer provided by Derek Holt in a sci.math thread:
A "standard" example of that is the Baumslag-Solitar group
G = < x,y | y x y^-1 = x^2 >
which is isomorphic to the multiplicative group generated by the 2x2 rational matrices
x = [ 1 1; 0 1 ], y = [ 2 0; 0 1 ],
and N = .
Then x N x^-1 = N and y N y^-1 < N, but N is not normal in G because y^-1 N y is not contained in N.
End of plagiarism.
Perhaps you can enhance this example to your purpose. Gerhard "Ask Me About System Design" Paseman, 2009.01.15