What were horrible problems that the Progressives had to solve?
The answer to this depends to some degree on how you define “horrible.” Were all the problems that the Progressives wanted to fight “horrible?” That is not something that can be answered objectively. One way to answer this is to say that the problems of impure food and drugs, harsh working conditions, and perhaps the lack of suffrage for women were “horrible” problems, though they were not the only problems that the Progressives had to or wanted to solve.
One “horrible” problem was impure food and drugs. Food companies could allow all sorts of disgusting things (as seen in Upton Sinclair’s The Jungle) to get into things like sausages that they sold. Drug companies could sell things that had no medicinal qualities and claim they were medicines. You can argue that it is horrible when these things happen.
You can also say that harsh working conditions were horrible. It is hard to argue that it was not horrible when a fire broke out at the Triangle Shirtwaist Company in New York City, killing 146 workers who were locked into the building. This happened in 1911. Not all harsh working conditions killed this many people, but many workers had to work in very dangerous conditions. Having to work in conditions where you can easily be hurt or killed could be seen as horrible.
Finally, we can argue that it is “horrible” that women did not get to vote. Voting is one of the most important rights in our democracy. We fought for independence partly because we hated being ruled by people for whom we could not vote. With that in mind, is it not horrible that half the population could not vote because of prejudice?
All of these were problems that the Progressives felt that they had to solve. It is not possible to objectively say that they were horrible, but it is certainly possible to argue that they were.