There seem to be many malfunctions, people getting duplicated, merged, and what not. Why do people still use the transporters? They don't exactly seem safe or reliable.
They kind of made a big deal in TNG6x02 Realm of Fear that transporters are officially "the safest way to travel". Even if they are incredibly dangerous, I'm sure you have reason to believe so, the other ways must be worse.
Also remember, often when they use a shuttle, they get abducted (TNG4x24 The Mind's Eye), crash on a barely habitable planet (TNG4x09 Final Mission and others) or get thrown out of space-time (TNG6x25 Timescape)? There seem to be a lot of accidents with shuttles. I would prefer the transporter, too.
Transporters ARE safe and reliable
People use transporters because they are the fastest and most convenient way to travel. In the 24th century, they are also considered the safest way to travel. The average rate of transporter failure was about one or two out of several million problem-free transports. That's any kind of accident period, not necessarily ones involving injury or death. Across the entire Federation.1
According to Geordi in "Realm of Fear":
"Transporting really is the safest way to travel."
We don't know how safe other methods like shuttlecraft are, but there isn't any indication that he was just giving his opinion. As an engineer he would have been basing that on actual stats. (unless you're on the Enterprise apparently; odds are all of the accidents that occur must be the few that happen on the Enterprise)
So although it's a good question when looking at things from our perspective, from the perspective of the average Federation man or woman, the premise is wrong: the transporter doesn't have many malfunctions. They likely almost never hear of any problems with transporters.
Some people DO avoid transporters
That said there are some people who agree with you and feel they'd better be safe than sorry. They sometimes even develop an irrational fear of the technology.
BARCLAY: Well, if I didn't know so much about these things, maybe they wouldn't scare me so much. I can still remember the day in Doctor Olafson's Transporter Theory class when he was talking about the body being converted into billions of kiloquads of data, zipping through subspace, and I realised there's no margin for error. One atom out of place and poof! You never come back. It's amazing people aren't lost all the time.1
Doctor's McCoy and Pulaski, actual medical doctors, were two more individuals who avoided scattering their molecules across space whenever possible. Their characterization is meant to be analogous to people afraid of flying on airplanes today. (I cannot agree on comparing it to automobiles as some comments have, since we're talking about the safety record of the technology itself, and most automobile accidents are due to operator error.)
The fear was strong enough in some people to be considered an illness. The EMH mentions "transporter phobia" in an episode of Voyager.
Going back further in time to right after the transporter's development, there was much debate about whether it caused health effects or other problems. So during that period many more people avoided the thing, but eventually the fear subsided as it proved safe (and became safer) over time. (Discussed in the Enterprise episode "Daedalus")
So there are certainly a small number of people who don't use them, but the convenience outweighs the infinitesimal safety risk for everyone else.
The same reason people fly despite the loss of life from plane crashes. It's convenient and probably safer than the alternative. Figure safe uses vs accidents. Despite the cases you've mentioned, I can think of only two fatalities, the folks in TMP. (The two redshirts in "A Child Shall Lead Them" don't count as they were beamed into space due to deceptive sensor readings, not a faulty transporter.)
If we are only comparing the transport methods of shuttlecraft and transporters, which are both mainly used for going planetside for away missions. However, shuttle crafts can also be used on longer sustainable trips like when Geordi was going on vacation to Risa (The Mind's Eye (Star Trek: The Next Generation)). I'm sure there are some technical specifications that would make a long distance transport through the transporters or possible industrial transporters not possible or it would create a larger margin of error that is unacceptable, but if we put that one aspect of the shuttle aside for the moment, let's look at the mechanics between transporting and using the shuttlecraft.
Transporters: From all the episodes that I have seen it is a solid-state system with little or no moving parts. Everything is handled by the transporter chief, computer and all of the 23rd century technology which doesn't seem to have any movable parts. I recall hearing that the subject is placed in a stasis field just prior to transport so that all of their atoms are in a slowed or stopped state for transport. Then they energize and they appear on the planet.
Shuttlecraft: Before anything a pilot proficient on that model of shuttlecraft must be assigned if one is actually available from their other duties. The away team must gather the shuttle bay and await the prepping of the shuttlecraft. Once everyone is aboard, the pilot performs all the necessary pre-flight checks and communicates with the hanger bay personnel to signal that they are preparing for departure. Hangar bay personnel activate the force field that keeps the atmosphere in the hangar bay, but allows the shuttle to pass through it and opens the hanger bay doors. Once given clearance, the pilot engages the engines to levitate and begin moving towards the hanger bay door opening. The shuttle then begins its course and heading toward an area on the surface.
Now...using a shuttle craft involves, a pilot, hanger personnel, back and forth communication with the hanger personnel during lift off (which during this maneuver human-error is in play. Unless of course the pilot is LtCmd Data). And if the shuttle clears the hanger doors and proceeds on course there is still the fact that you are flying a shuttle from a 0 atmosphere to a possibly unknown atmosphere and possibly unknown meteorological phenomenon that could either damage the shuttle or catch the pilot unaware and unable to respond properly in time endangering the pilot and the entire away team.
So, in summary, I see the comparison between transporters and shuttlecraft the same difference between a hard disk drive and a Solid-state Drive (SSD). The first has moving parts, the second does not. (Also, the SSD is much, much faster than the spinning disks that will eventually wear their bearings out because of the basic force of friction, regardless of how little friction it will eventually fail.) HDD usually have on the box a characteristic called MTBF. This stand for Mean Time Between Failure. Usually it is calculated in hours, but averages about 5 years.
I hope that I've made the simile clear. Please let me know if there is an error or you need fact checking or references added.
This is my first post & I'm not referencing any of the canon of Star-Trek, merely making a logical comparison between the two modes of transportation.
I have to challenge the whole premise of your question.
There seem to be many malfunctions, people getting duplicated, merged, and what not.
In TNG6x02 Realm of Fear, an experienced and respected Starfleet engineer flat out states that transporters are "the safest way to travel". There aren't that many malfunctions in the usual day-to-day happenings across the United Federation of Planets or any of the other transporter-using cultures. The incidents we see on film in the assorted shows, movies, and whatnot are outliers.
Because Star Trek is presented to the viewing audience as entertainment, we only see many outliers, because outliers are interesting and exciting. If it weren't an outlier, it probably wouldn't be worth showing on screen. I'm not inclined to rewatch every episode and count scenes, but I suspect if one were to count up all the instances of transportation (including those that are implied but not shown), the interesting incidents might still be outliers.
While it is presented tongue-in-cheek, because it's also meant as entertainment, consider Chief O'Brien at Work for a more "realistic" (used loosely) look at how dreadfully boring transporter operations probably are.