How do you make a balsa bridge stronger without making it heavier?

Short answer: stop spending weight where nothing fails, and fix the thing that does. Most gains come from better joints, shorter compression members and better stick selection, not from more wood.

1. Find the first failure. Test a practice bridge and note what broke first. Fix that, not everything. See why bridges fail.
2. Shorten compression members. A stick that is pushed buckles, and doubling its free length cuts its buckling load to about a quarter. Add bracing or another panel.
3. Sort your sticks. Weigh each one. Use the heavier, stiffer ones where sticks are pushed, and the lighter ones where they are pulled. See choosing sticks.
4. Fit joints tightly. A joint with no gap is much stronger than one with a gap. Sand until no light shows through.
5. Use less glue. A few milligrams per joint. A coach's rule of thumb is glue under 5% of the bridge mass. See gluing balsa joints.
6. Taper the build. Put more wood in the middle of the span, where forces are greatest, and less near the supports.
7. Brace sideways. Cross-members between the two trusses stop it twisting before the sticks give out.

Change one thing per version, so you know what helped. See testing your bridge.