How does a new species survive without suffering inbreeding? Just what the title states. The thought came from reading When has an organism evolved enough to be called a new species? I'm probably wrong but I understand new species happen sporadically rather than in whole-sale lot during breeding season. Is the new species capable of breeding with members of its parent species? How does a new species survive inbreeding initially?

Well, speciation is a process, not a single event thing.

Usually what happens initially is geographic isolation of some members of a species, so in that sense the inbreeding is something that is influencing and you can also think of it as causing the speciation process. The presence of special harmless or semi-harmless disadvantagious mutations - which increases in frequency from this isolation - influences how the species is going to evolve, but eventually shapes this species to better cope adapt to some environment, or evolves the new species out of existence.