What causes the issue is staring at a very bright background continuously. This causes the cells at your retina to get tired, which makes reading almost impossible.

There are two fixes:

1. Avoid using large very bright backgrounds (duh!). Any large surface should be something other than bright white.
2. Variation. The issue is not really caused by something being too bright. (As you said you used the same white.) It is caused by the users only looking at something with a bright background all the time. There should be darker surfaces people look at, so the eyes can rest. Large images, titles with colored backgrounds, sidebars... doesn't really matter what as long as it is something people spend time looking at, so that they do not spend all their time looking at something bright. Optimal for reading might be making the font very large, as that would give your retina lots of variation and proper average brightness, but it isn't really usually practical.

What causes the issue is staring at a very bright background continuously. This causes the cells at your retina to get tired, which makes reading almost impossible.

There are two fixes:

1. Avoid using large very bright backgrounds (duh!). Any large surface should be something other than bright white.
2. Variation. The issue is not really caused by something being too bright. (As you said you used the same white.) It is caused by the users only looking at something with a bright background all the time. There should be darker surfaces people look at, so the eyes can rest. Large images, titles with colored backgrounds, sidebars... doesn't really matter what as long as it is something people spend time looking at, so that they do not spend all their time looking at something bright. Optimal for reading might be making the font very large, as that would give your retina lots of variation and proper average brightness, but it isn't really usually practical. You can also give the background subtle low contrast texture.