At the first general election after the adoption of township organization by a county, one supervisor shall be elected in each supervisor district. Thereafter one supervisor shall be elected in each odd-numbered supervisor district at the general election two years after the first general election and each four years thereafter, and one supervisor shall be elected in each even-numbered supervisor district at the general election four years after the first general election and each four years thereafter. Each county supervisor shall be nominated and elected by the registered voters of the district from which he or she is elected. Except for supervisors first elected after the county has adopted township organization, the term of each county supervisor shall be four years or until his or her successor is elected and qualified. The county supervisors shall meet the qualifications found in section 23-268. The county supervisors shall be elected on the partisan ballot.