Abstract: We present a unified likelihood ratio-based confidence sequence (CS) for *any* generalized linear models (GLMs) that is guaranteed to be convex and numerically tight. We show that this is either on par or improves upon known CSs for various GLMs, including Gaussian, Bernoulli, and Poisson. In particular, for the first time, our CS for Bernoulli has a $\mathrm{poly}(S)$-free radius where S is the norm of the unknown parameter. Its derivation follows from time-uniform PAC-Bayesian bound with *uniform* prior/posterior, which is our main novelty despite being a rather unpopular choice for deriving CSs. As a direct application of our new CS, we propose a simple and natural optimistic algorithm called GLM-UCB+ applicable to *any* generalized linear bandits (**GLB**; Filippi et al. (2010)). Our new analysis shows that the celebrated optimistic approach attains a regret whose leading term is completely free from $e^S$ for self-concordant (not necessarily bounded) **GLB**, and even $\mathrm{poly}(S)$-free for bounded GLMs. Especially for logistic bandits, we show that GLM-UCB+ can attain even better regret bound, and we verify numerically that GLM-UCB+ outperforms the prior state-of-the-art (Lee et al., 2024).