You should describe the format of your polygon data, to avoid us guessing.

In general many efficient so called "clipping" algorithms depend on calculating "bounding boxes".

This - the smallest surrounding rectangular - allows eliminating many impossible candidates.

The rules are:

• Bounding boxes don't intersect => Polygons don't intersect
• Bounding boxes can only intersect if both x and y intervals intersect.
• Two intervals a (= [a0,a1] ) and b only intersect iff a0 <= b1 and b0 <= a1

Thus storing the bounding-boxes in an ordered structure° helps efficiently eliminating impossible combinations.

But I suppose you are better off using an already available graphic lib.

HTH! :)

update

°) see also Spatial_index#Spatial_index and for instance Quadtree