This looks like an "alpha clipping" issue.
Your planes look like they have an image texture applied to them. If this is correct and the image has an alpha channel (for transparency) then try setting the Material Properties > Viewport Display > Blend mode to Opaque.
Example Plane textured with an Image with an Alpha channel:
Or even better Open the image in Gimp or Photoshop and remove the Alpha Channel ( Gimp : Layer > Transparency > Remove Alpha Channel
Edit to add a couple more screenshots after Hienronymous's comment.
The Show Backface option is not available when Blend Mode is set to Opaque but there when Blend Mode is set to the default Alpha Blend
Which (probably) makes sense since the two seem to preform similar functions.
The Blend Mode Opaque "Renders surface without transparency" and with Show Backface deselected "avoids transparency sorting problems".