How does a new species survive without suffering inbreeding?
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.