Thanks everybody, but I'd appreciate an advice - what to do?
Should I call my insurance and let them take care of the bike repair, or
should I claim with the city? and do not call insurance? Or both?
Regardless of whether the City accepts the claim or not, insurance will most probably treat this as an at-fault collision, with the rationale being that you should adjust your riding to suit road conditions. They will point to the left lane as being where you should have gone.