Is a bounded and continuous function uniformly continuous?

You're close: $$\sin\frac{1}{x+1}$$ is a counterexample to the statement.

For continuity to lead to uniform continuity, domain has to be compact, and as you can see the domain is not compact here. Also, rightly $f(x)=\sin(\frac{1}{x+1}) $ serves as a counterexample or even $ \sin(e^x)$ for that matter.

$\sin(x^2)$ is also a nice example and it's happening because it's not periodic.