How can many Jews still believe in God after being persecuted and killed? Why are they faithful to a God who did not protect them?

That is the way religion works. The problem with it is that people are not reasoned into religion. So they are extremely difficult or impossible to reason out of it.

No matter how bad things get, and I cannot imagine worse than the Holocaust, people still believe in their gods.

Of course no gods exist and the god of the Jews did not abandon them to their fate. The Holocaust was the result of the Nazis and their sympathisers.

It should, of course, make rational people think. Why would god let his chosen people suffer such horror on so great a scale? But, as I said earlier reason does not come into it. Therefore, despite this great horror Jews do not abandon their god as he seemingly abandoned them.

Their profound innate superstition overrides rational and logical thinking..