We often say that a zoysia lawn from seed will take 10- 12 weeks to develop a lawn appearance and be suitable for light use, or even up to 16 weeks, depending on the weather and time of year it is sown, and the degree of agronomic intervention.
It is certainly hard to force a zoysia turf to grow a lot faster - more nitrogen fertiliser just does not work. After all, that aspect, of slower and maybe even less growth is one of the benefits of a zoysia turf area once established, and these turf areas can be used for many years. My own zoysia lawn at home is now about 35 years young........and still doing okay.
In the world of contracting, a period of 16 - 20 weeks is considered a common period for the development of a large turf area eg a playing field, from seed, suitable for sport use. This is a useful concept to understand.........it will not happen instantaneously [ unless you use full sod at much increased cost - many times the cost of seed establishment!].
And there can be a few trials and false starts along the way.
Our common recommendation is to mow off the weeds, especially before they set seed. The zoysia continues to grow laterally close to the ground, filling in the space, even when a bit shaded by the weeds, and will normally choke out most weeds. In warm regions, summer growing grasses are vigorous, and sometimes using a wick wiper treatment of glyphosate carefully can kill them off - but avoid getting herbicide on the zoysia - it is susceptible to glyphosate.