How can we play yu-gi-oh online after dueling network's shutdown?
Building off of DarkCygnus' answer, the "spiritual successor" to Dueling Network is Dueling Book. It is stylized in the way Dueling Network used to be, has both a competitive ladder and casual matchmaking, and is a "manual" simulator (you have to resolve the effects of cards yourself, place them in the zones they should be in, etc.) The other simulators in DarkCygnus' solution are "automatic" simulators, that give you prompts for things like activation windows for cards and proper zones monsters can be summoned into.

If you are looking for something similar to Dueling Network, Dueling Book would be my recommendation.