You didn't specify cost of glue. If the glue is of no cost (other than end to end joints as you mentioned) then my inclination is to think outside the box here.
You could shred the toothpicks into fibers (pull them apart, grind them up, whatever) and then use the glue to create a composite matrix using the fibers.
You could get a lot of stringy fibers from very few toothpicks.
I don't know all the rules, but If you can bring stuff in other than just a hairdryer, then I would bring in a food saver type vacuum bagger and some 10 or 20 lbs foam, and some Vaseline.
Carve a negative form for the bridge into the foam and coat it with vaseline (as a mold release agent). Mix up the wood fibers and glue and spread this mixture out into the form. Slip the whole thing into a food saver bag and turn on the vacuum pump to apply pressure while it cures.
Use the hair dryer to apply heat over the whole thing to help cure while it's in the bag.
Finally, once your WFRG (wood-fiber-reinforced-glue) bridge is cured you would remove the whole thing from the bag, and pop the bridge out of the mold. Done.
I didn't see time limit listed in your question, but I would imagine all of this would be way faster than building a more traditional bridge gluing individual toothpicks together, so I assume the time allotment would be sufficient.
Again, depending on the rules about bringing stuff in, you might be able to make the mold ahead of time and bring it with you, so all you have to do during the construction phase is shred wood, mix with glue, apply, and cure.
Something I learned at an early age is it's better to ask forgiveness than ask permission. If the rules don't specifically ban you from bringing in this outside stuff then I would say it's fair game.
I believe this fiber-reinforced-matrix construction could work well. However I've never done it myself, so I'm not positive. It's similar in concept to fiberglass or carbon fiber, GRFC, or FRP, and conceptually would work the same way and offer the same benefits.
Traditionally, fiber-reinforced products work because the fibers are suitably strong (overall, not individually) to resist great tensile loads, while the matrix binder (in this case the glue) provides adequate support to hold the fibers in place.
As it's your homework assignment, I would do your research on FRP's (fiber reinforced plastics) and other composite materials to determine whether or not this will indeed work for you.
An H-Beam or I-Beam form I think would prove suitable.
You could also look at adding gussets and ribbing within your foam mold to increase strength and rigidity at areas of stress concentration.
Looking at the rules you've written, I don't see how this would violate any of them.
If you have access to 3d modeling software and FEA software at your school, you could do an optimization study to give you a good idea of how to build the bridge and where to add gussets to assure structural integrity. Many FEA suites offer the ability to apply composite materials.
Also you can look at the way many plastic playsets and slides are constructed. These are typically roto-molded plastic, not composites, but the beam construction and ribbing could give you some hints as to the approach to take.
You're mileage may vary, but I'll tell you this - if I was a teacher, who did this project every year and watched countless students create countless iterations of the same basic bridge design using girder construction, I would be pretty impressed if someone brought in a completely fresh approach to the problem.
This attitude of thinking differently has served me well in my career, it may or may not work for you. Just a thought.