Spiders spin silk almost like silkworms. What they just spit out is mucus. As soon as it touches air, it turns into silk.

Spiders build bridges by spitting out silk long enough to let the wind drift to the trees on the other side. Spiders come to test with their feet. If they find that they can't pull it, it means that the silk has been hung on the opposite side. As long as there is a hanging silk, the spider can climb up and run back and forth to spin silk, pull out the main silk and then make a web.

Spiders build bridges not to cross the river, but to catch small insects flying around the river for food.