The thalidomide scandal is an example of what you are speaking about above, but that happened in the 1950s. Regulations for approving drugs and vaccines are far stricter today than decades ago. People are naturally sceptical because of the speed with which they were developed but there were also huge sums of money poured into the research, and its creation was expedited more than any vaccine in history because it was a public health emergency.
Also, you can also make the same case for having the vaccine, because no one knows what the long term effects are for people who have had the virus, especially for those with "long Covid", for whom there can be serious debilitating mental and physical effects even in the short term. Only time will tell for either stance.
There are things you can get from a vaccine such as blood clots or myocarditis, which, like effects from the virus, may be underlying and exacerbated, but the numbers are so tiny (19 cases from 20 million vaccinations for myocarditis with Pfizer or Moderna) that I personally read them as almost negligible. Covid itself can damage the body in lots of ways - lungs, brain, heart and multiple other organs, plus arterial issues, causing blood clots, strokes, heart attacks and permanent lung damage, even (in lower numbers) with young people. There is a 31 year old goalkeeper from Birmingham City who is currently in hospital with it. Obviously there is far greater risk for older or weaker people, but the numbers don't balance the scales in favour of not taking it.