When I first moved into my place 9 years ago, the lawn had a few weeds in the dandelions....weeds such as Fescue, ryegrass, bluegrass!!!
After the first year of mowing at 3.5 to 4 inches high, most of the dandies were gone and the yard thickened up. I still spot spray in the spring and fall with 2-4,d Amine 400 from TSC to keep them away.
Also, fertilizing the lawn regularly will keep white clover and dandelions at bay as they generally thrive in low nitrogen, high PH soils where there is less competition from grass.
Brian