This is a common problem for manhattan plots. An easy way around this issue is to just subset p-values below 0.05 and only plot those. This will give you a simple illustration of the important results you can work with in the meantime. When it comes to making a publication quality plot, you can subset a few thousand random p-values above 0.05 per chromosome and this will fill in the background. The vast majority of the non-significant data points will overlap so they become redundant very quickly.

There is a note about filtering and making nice plots in RGraphGallery's page on Manhattan Plots.