If the west side's water situation is unsolvable then they might put defenses on the west side and use it for control of the strait and as a bolt hole in response to an attack from the steppes. The swampy lands might be enough to slow attackers enough to evacuate across the strait. There would also have to be enough defense in depth to allow evacuation.
So, people would live and farm on the east side. The military and, likely, rulers would live on the West side. If the granaries are on the west side, the drier climate would be better for storage. I would also put the religious institutions on the west side to reinforce the fact that the townsfolk need the west side.
If the people can evacuate across the water, steppes attackers would go through all that effort for nothing. Also, if the military gets good at water navigation, they can probably put harassing troops behind the attackers for hit and run attacks. All this is an effort to discourage further attacks. The combination of being a pain in the rear to attack and having food for trade might be enough to work out a peaceful arrangement with the nearest steppes tribe.
If the steppes tribes are too much of a threat, then most of the population can be on the west side. Then water may have to be transported up to the city. In this scenario, you only have farmers on the east side. The evacuation would be quicker that way but more effort needs to be put into water and food transport.
I just looked at the map on Google. It shows water sources and farming on the west side. It might not be as good as on the east side but, combined with farming on the east side, I think the west side can support a decent sized city. I don't get a good feel for relative elevations so I don't know how difficult it would be to get water to the city from the small lakes up there. So, city on the West and farming villages on the east sounds like a good solution if you want grain for trade.