It is possible to sue a government entity (such as a city, school district or state) for causing my injury?

In the case of a car wreck caused by a government employee, Yes, you may file suit against the City, district, county or state.  However, there are strict (and short) notice requirements as well as other limitations.  In the case of injuries caused in other ways such as negligently maintained buildings or roads, filing a claim is allowed, but only in limited situations.