The fastglm package does not compromise computational stability for speed. In fact, for many situations where glm and even glm2 do not converge, fastglm does converge. As an example, consider the following data scenario, where the response distribution is (mildly) misspecified, but the link function is quite badly misspecified.
1: glm.fit: algorithm did not converge
2: glm.fit: fitted probabilities numerically 0 or 1 occurred