This sounds like a "go around". When an aircraft is landing the sequence is very dependent on a number of factors, which includes the speed of the aircraft in front of you and the time it takes for the plane ahead to exit the runway. If the aircraft ahead of you lands long for example and misses the first exit of the runway, the delay in taxiing could cause the aircraft behind him to "go around". Because the aircraft is configured for landing, the "go around" procedure has a single button which automatically spools up to maximum power, adjusts the flaps, etc.. The effect is dramatic and feels uncomfortable (I have been through it a few times), but it is absolutely not unsafe.
It is equally possible that your hypothesis that the pilot had a "missed approach" is true and had to "go around" but this is also not unsafe. I think you have had an experience which has made you anxious, but I think it is highly unlikely that anything untoward or unsafe was taking place. Incidently, if it did, the airline will know about it and will check. "Go arounds" are reportable incidents and the airline will examine what happened. If the pilot was on an unstable approach, he will certainly be under scrutiny.
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