Are ibuprofen side effects something to worry about?
Short answer: maybe.
Longer answer: As with a lot of health questions, it depends on how they are being taken. If, for example, you hurt yourself playing a sport and want to take a painkiller for some short-term relief, ibuprofen probably won't do you much harm in the short term.
On the other hand, if you are in chronic pain (or inflammation, as another answer notes) the situation is a little different. If you are taking an NSAID - ibuprofen is one of several in this class of drug - on a regular basis it's worthwhile watching out for some side effects. One mentioned in the article you linked are stomach ulcers as NSAIDs can have an effect on the protective lining of the stomach. Usually some form of 'gastroprotection' is offered to offset this, such as a PPI (proton pump inhibitor, such as omeprazole).
As another example, if you are taking ibuprofen for recurrect headaches (eg migraines), you can counter-intuitively end up getting another kind of headache, called a [medication overuse headache(https://www.migrainetrust.org/about-migraine/types-of-migraine/other-headache-disorders/medication-overuse-headache/) (MOH).
The bottom line is that any long-term use should be discussed with a medical professional; who can assess relative risks and check for interactions with any other regular medications.
But ibuprofen/NSAIDs can't be safe! Vioxx was taken off the market...
Quite true. From the article you quoted:
In the early 2000s, a number of large studies found a significant association of negative heart events, such as heart attack and stroke, with the use of selective COX-2 inhibitors. This resulted in two of these drugs, Valdecoxib and Rofecoxib or Vioxx, being withdrawn from the market.
The Vioxx controversy was a pretty big deal, and resulted in compensatory payouts and fines. The VIGOR study demonstrated a 4-fold increase in heart attack risk on rofecoxib (Vioxx) versus naproxen. This was attributed to the 'cardioprotective' effect of naproxen. This was incorrect, and another trial (APPROVe was stopped early because selective COX-2 inhibitors demonstrated adverse cardiovascular events. There was a raft of litigation relating to the sale of Vioxx, with payouts and fines in the billions of dollars.
If you are interested, the Vioxx controversy created lots of discussion ongoing to this day- from editorials, to book sections, entire books themselves and arguments relating to tort reform. It is clearly beyond the scope of this question!
That said, not all NSAIDs are Vioxx; and as noted above, it is possible to take them (even ibuprofen) relatively safely.
What are the alternatives to ibuprofen?
As another answer comments, it depends on what you are taking ibuprofen for. As mentioned above, you may be fine continuing to take ibuprofen, depending on circumstances.
- naproxen (another NSAID, but with lesser risk / may be protective)
- paracetamol / acetaminophen (ie Tylenol, as you say)
- NSAID cream / gel, eg for joint pain
- tramadol may help for some kinds of pain, but there are issues with addiction and constipation
- alternatives, such as exercise, acupuncture, meditation etc can all potentially reduce pain