Just use percentage based layouts rather than pixels or other measurements.
<img width="50%">: that will fill half of the screen, at any size
<img width="500px">: that will always fill exactly 500 pixels, if it's too big or if it's too small.
It is also theoretically possible to design a website using pixels and adjust it using Javascript to scale, but that is the same as using percentages (in how it looks) and MUCH harder to do. It also is unreliable. Basically, a good web designer won't need to do that.