There are a few nestled misconceptions that have to be cleared up here. I will get to your question at the end, but it's important to understand the sequence of events that lead to the Nazis actually coming into power.

I ask this because he won in a democratic election.

Not quite! Hitler had first tried to run for the post of president (which carried substantial executive powers with it) in 1932, but lost to von Hindenburg. However, he did well enough to force the establishment to take him seriously. The NSDAP had earlier in the year, in the July election, gotten a plurality with 37% of the vote. However, it lacked support to form a majority of its own.

However, the system at the time in Germany was in practice not very democratic, thanks to the extensive powers invested in the presidency. von Hindenburg had opted to let Franz von Papen and his Center party, with only 14% of the vote, to continue to run a minority government. This did not last and led to a new election in November the same year. The Nazis did not do quite as well here, getting only 33%, but together with Hitler's relatively good performance in the presidential election and the unsustainability of keeping the Centre Party in power was clear to everyone involved.

At this point, the establishment politicians (Hindenburg, von Papen, etc.) had basically three options. One, they could try to form a coalition with the socialists and communists plus some minor parties to gain a majority - this wasn't going to happen for multiple reasons as it was politically unacceptable to both the Communists and the Centre Party. Two, Hindenburg could declare martial law and essentially install a military dictatorship. Three, they could form a coalition with the Nazi party plus nationalist parties.

Not wanting to lose his own influence in politics, von Papen approached Hitler in January 1933 and told him that he held enough sway with von Hindenburg to convince him to appoint Hitler chancellor, although it required significant concessions from Hitler, allowing him only two ministers (Wilhelm Frick as Minister of the Interior, and Hermann Göring as Minister without Portfolio). Hitler's position as Reichskanzler was not thought to be a particularly important one in the day. The plan of von Papen's was to "box Hitler in" by retaining control of all important ministries. This might have been possible, if not for the big watershed moment: The Reichstag fire.

Following the fire on February 27 1933, which was blamed on the Communists, the NSDAP drafted an emergency decree for von Hindenburg to sign (the Reichstagsbrandsverordnung) which granted the cabinet (in practice, Hitler) sweeping powers and the ability to suspend civil liberties. This allowed Hitler to exercise power in a very direct, physical manner - by having his men take over police stations across the country, and by legitimizing his own paramiliatries, the Sturmabteilung, as an auxilliary police force. He was able to arrest Communists for being Communists, install infrastructure for the mass distribution of propaganda in Berlin, and have the NSDAP infiltrate the German state on every level. The extent of this does not appear to have been apparent to von Papen and other establishment politicians at the time. Moreover, on March 5, a snap election was held, "monitored" by the NSDAP's own paramilitaries, where they were able to secure a bigger share of the vote (43%).

The next big blow comes when Hitler, having arrested every Communist (and some Social Democratic) representative in the Reichstag, puts up an Enabling Act for vote, a constitutional amendment that would grant him powers so sweeping as to practically void the Weimar constitution. The Nazis are able to pass this with support from the Centre Party (who do not appear to have realized what was happening and believed they would get significant concessions and guarantees from the NSDAP). The vote was of course "monitored" by the NSDAP:s paramilitaries.

Following the passing of the Enabling Act, Hitler promptly dissolved the Reichstag. On July 14, he banned all parties except the NSDAP. At this point, the only real threat to Hitler was the octogenarian von Hindenburg, who was suffering from dementia and was largely being carted around as a prop by the Nazis. In priniciple, he could at any point have declared martial law and had Hitler deposed, which he did threaten to do in 1934, although Hitler appeased him and the army through a purge of his own paramilitaries. On von Hindenburg's death in August, Hitler declared himself Führer, formally assuming the powers of the presidency as well.

As you can see, there was a very rapid sequence of events in 1933 following which it would be apparent to everyone that the Nazi government was highly anomalous. How did they feel about this? Well, one of the most interesting works on this is Kershaw's The Hitler Myth. Kershaw argues convincingly that Hitler was quite popular, whereas the NSDAP and the Sturmabteilung (known for their clashes with the Communists in the streets) were not. The worries most people expressed were about Hitler not having enough control over his subordinates, rather than Hitler having too much power.

It's important to understand that the Weimar republic was not only a young democracy, it was an extremely dysfunctional democracy. Elections on elections on elections were held, cabinets with ridiculously tiny minorities were appointed through back-door deals, laws were passed as emergency decrees by the president, and the Reichstag was chaotic. It is therefore understandable tht people weren't concerned so much about one person monopolizing power, but about having a functional government at all.

Sources and further reading:

The Hitler Myth (Kershaw)

Hitler: Hubris (Kershaw)

The Nazi Dictatorship: Problems and Perspectives of Interpretation (Kershaw)

Third Reich Trilogy (Evans)