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let us assume that we have a conductor with a specified resistance ( case 1 ) and a normal conductor ( case 2 ) as shown in the figure , and now we apply an external electric field with a battery on it, we know that current density will be different in both the parts ,however , both the parts of the wire have different resistances as there is a change in its cross section area but the potential difference will be equal as the length of both the parts is equal so is it correct to say that current running through both the parts is equal ?answer for both the cases enter image description here

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The potential difference across them wouldn't be equal because their lengths are equal. It would only be equal if their resistances were equal, which as you point out they are not.

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If they are in series, then yes the current will be the same flowing through each, it has no other path to take.

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