While you say the bits were cheap and you think they're to blame1 there's a good chance your collet is shot also, because once bits become loose under speed they will tend to wear the collet too as well as their own shaft. Once the collet is worn you get an effect call runout where the bit can deflect from the rotational axis. Runout causes vibration and chatter, and of course at worst the bit can work free during use.
Source: The New Router Handbook by Patrick Spielman.
But there is a chance your collet hasn't worn excessively and just needs to be cleaned well so that it can tighten properly. Sticky wood dust and resin can build up between the fingers of the collet (even inside the hole) and it needs to be cleaned away using an appropriate solvent and small brushes.
Prevention is not to be taken literally here since some wear is inevitable, but to lessen wear in future it's important to follow good practice when installing a bit2. Not sure if it's the main thing but it's important never to tighten the collet nut with a bit fully inserted or "bottomed out", always back it up slightly.
The bottom edge of the cutter head on the bit (or the bearing if it has one there) should never press against the rim of the collet, but there's also this which isn't as obvious:
Source: Wonkee Donkey Tools.
As a quick visual reference for installing bits to their proper depth in the collet it's worth marking their shafts with a sharpie after you've determined the ideal position, the line will instantly tell you that's in far enough and not to mount the bit any deeper. This line will probably need to be renewed periodically as cleaning resin from the bit will eventually wear it away.
1Cheap bits are a potential problem here because reportedly their shafts can be undersize (by a very tiny amount, but it's enough) which in addition to allowing for play in the bit when spun up can also stress the collet if it's tightened down enough to grip firmly — overtightening collets can lead to premature failure.
2Your router's manual probably has all the basics laid out in the introductory pages and it would be worth going over them again to commit them to memory, then doing some practice drills until good installation practice is second nature. Can't be too safe.
If you don't have the paper manual for your router an electronic version will be available for download online (this is a legal requirement everywhere I believe).