fighting a lingering conflict with Algeria's security forces

Local officials say that Islamist insurgents, fighting a lingering conflict with Algeria’s security forces, are targeting the town’s small army of unemployed young men and persuading some of them to join their ranks.
I want to rewrite the bold phrase as follows.
“fighting with Algeria’s security forces in a lingering conflict”
Is it better than the original?
I feel that the original isn’t easy to understand. I don’t see the structure “fight something with something” in my dictionary.

The original is fine and there is no reason to rewrite it.

Thank you, Dozy:
The structure is a bit difficult for me to understand. I have consulted Longman English Dictionary about the structure but I still found no results.
I found the following:
fight with:
Two guys were fighting with each other in the street.
Therefore, I think it is correct to change the phrase above into “fighting with Algeria’s security forces in a lingering conflict” or " fighting in a lingering conflict with Algeria’s security forces".

It does not sound fluent. A native speaker would not use the preposition ‘in’.
In the example you found, ‘in the street’, the preposition ‘in’ is used to denote the place where the fighting occurred.

It does not sound fluent. A native speaker would not use the preposition ‘in’.
In the example you found, ‘in the street’, the preposition ‘in’ is used to denote the place where the fighting occurred.

“fight” can use a direct object in two different ways. In the first way, the object is the opponent, as in “Tyson fought Holyfield twice” or “I fought the law”. In the second way, the object is the conflict, as in “We fought a great battle”. The phrase “fighting a lingering conflict with Algeria’s security forces” is a combination of this second object pattern, plus use of “with” as in your example “two guys were fighting with each other”.

Thank you Beeesneees and Dozy!

This is exactly what I am confused.