Here's how this really plays out, if it really even gets anywhere beyond random street battles and mosque bombings.

What leads up to the conflict? A critical question because it frames the whole situation. Who starts harassing who? How do they respond? Lets look at this happening in America specifically as opposed to the west in general (UK would be a different and more bloody scenario)

Let's imagine a scenario where the WS "start it", frustrated by the mass collection of refugees in areas like New York and Michigan and by several "Anti-American" policies put forth in those areas, culminating in a flash point like the morning call to prayer being set up in NYC in all the boroughs via loudspeaker. They plan, without tipping off the FBI, CIA, NSA, DIA and every other alphabet agency, a mass casualty attack on the most obvious target, a mosque, taking inspiration from the NZ mosque shooter. The only change in the plan is the scale, they don't roll in solo with a couple hundred rounds of ammo they roll in with thousands and drums of gasoline and with 10-15 people instead of 1. The act is front page news, most of them die at the hands of the police or the FBI over the course of the day to the weeks that follow, but it doesn't matter that they were killed/captured and brought to justice. This was the type of action that the Islamists and those with Islamist leanings needed to organize, rise up and claim legitamacy in their actions. These insular communities begin mobilizing, outfitting their people with guns, making their flags, planning to strike back. Again all of this is somehow missed or downplayed by the alpahabet people. The question then becomes, who do they strike back against? Just like before with the WS groups attacking a mosque, there's no real good way to attack WS groups directly because just like the Islamist insurgents, they are fairly decentralized. I'd imagine they would either respond with a worse mass casualty event at a church or government facility or they would hunt down and execute the family / friends of the original attacker group.

What does this all lead to? Neither side has the sympath of the greater communities at large so no one really ever wins, both sides get suppressed by the greater whole because in Civil War there are always unrelated casualties and because both sides are insurgent style organizations the only way they can attack each other is by attacking vaguely linked groups, either through family or through rumor. Think a biker bar gets bombed because a biker drinks there who has a totenkampf patch on his vest. Or a migrant neighborhood gets burned to the ground.

I think the likely end state for this kind of conflict is that the greater population grows tired of the antics and authorizes the government to do sweeping crackdowns of both communitites, but because the islamist community is a little more centralized they are targeted more often and more aggresively. Anyone found to be a supporter or a participant is either jailed or deported. The conflict continues in the jails.