From the wackypedia article,
There was mass panic as the blaze jumped the river’s main stem and continued burning through homes and mansions on the city’s north side
How does a fire ‘jump’ the river? Granted I dont know how wide the Chicago river is, but could have really been lava-like hot like some kind of Hiroshima bomb? How could the fire not be stopped by the river? I don’t get it
My best guess would be blowing debris and ashes from the structures on one side of the river being carried by the breeze to the structures on the other side.
Wind and embers.
The river in Peshtigo (which burned at the same time and was far worse in terms of death toll and damage) didn’t stop that fire. The wind had turned the fire into a firestorm, which blew burning embers across the river and coals across that river.
When embers from burning material was blown over the river, the other side caught fire; this kind of thing will happen all the time, but usually not enough embers reach the other side to start a blaze on the new side. In the case of the Chicago fire, there were enough (in some places the river was narrow enough and the fire hot enough that embers weren’t even needed - the fire heated material on the far side enough that they simply burst into flames)
Anything’s possible with a running start. 
Worth noting that the fire had to jump the river at least twice - from west to east relatively early on and later from south to north in order to reach the north side. For the latter there were stories of people seeing windborne burning debris land on a railway wagon carrying a tank of kerosine, with explosive results.
There were also reports that there was sufficient oil pollution in the water that the river itself was on fire in places.
According to the map in the Wikipedia article, it looks like the fire jumped 20 miles across Green Bay.
A couple of other points: One, a very large fire (like the better part of a city burning) will itself create very strong winds which will tend to spread burning material far. Two, the Chicago river isn’t really all that wide, anyway. Between those two points, it’s not at all hard to imagine burning bits of paper or thatch or other lightweight materials blowing across the river.
The howling strong southwest wind was throwing firebrands hundreds of feet out in front of the burning buildings. Those then started subsidiary fires, which burned back to join the flame front. Here’s a map I drew to illustrate the progress of the Fire, based on a careful contemporary reconstruction.
Wow, very nice!
Here’s a picture of the Chicago River as it flows through downtown Chicago. Of course, the banks of the river weren’t nearly so built-up in 1871, but you can still see that the river is not wide at all.
That is the conventional wisdom, but many historians now say it didn’t happen, but fires started independently on both sides.
I’m not doubting that it could happen.
Here’s a video lecture about the Peshtigo Fire told by Barb Chisholm in period costume.
We had a block of apartments under construction here catch fire and burn to the ground over a weekend and the radiated heat charred hedges and fences on the other side of a broad road and melted uPVC window frames and gutters on the houses there. Those houses might well have caught fire just from that - and it was only one building.
