The conjecture is false. For example, consider the dihedral group G = D8 of order 8.
D8 = <r,s | r4=s2=1, rs =sr-1>
This group has two subgroups of order 4, namely N1=<r> and N2=<s,r2>. These subgroups are normal since they have index 2. The quotient groups G/N1 and G/N2 are both isomorphic to Z2, but note that N1 and N2 are not isomorphic (one is cyclic, the other is not).