Drill a hole in the "top" piece of wood roughly the diameter of the screw. That way the screw's threads can't bite into the top piece of push the two pieces apart.

There are many kinds of screws. Screws designed to be used in wood have a smooth top portion on the shank(the threaded part of the screw. That smooth portion prevents the very problem you experience.

The link below illustrated that difference.

Here is a link that might be useful: Screw tutorial

The un-threaded shank should be as long as the top piece is thick.

For classic tapered wood screws, Since this is not always possible, you drill a larger shank size hole trough the first piece of wood, then a smaller pilot sized hole into the second.

For non-tapered screws (or screws with no bare shank) the hole through the top piece is the diameter of the threads so they do NOT grab in that piece.

The pieces should be clamped together while driving the screws.
Be sure to use wood clamps or some scrap smooth wood padding to avoid clamp marks. You just need the clamps snug, not muscled down to the n-th degree.

There are drill bits that are tapered fr classic screws, and have a step for non-tapered screws if you have a lot of holes to drill.

A flag of masking tape around the bit can be used as a depth stop.

if you are using brass screws, always run steel screws of the same size in almost to tight to cut in threads.
Bras screws are soft, and especially in hardwoods can break off trying to cut in thread s in a fresh hole.

I keep at east one matching steel screw in every box of brass screws.