Why does stretch fabric keep getting stuck in needle plate?

SOURCE: needle hits needle plate with heavy fabric

that is your problem, your needle bar is centered too far to the back. There are 2 little bolts (Below the presser foot lift on the back of the machine) that must be loosened while the needle bar is repositioned aligning the needle bar assembly.

SOURCE: I WAS SEWING THROUGH SEVERAL LAYERS OF FABRIC MY

Refer to the manual. If there isn't any information there. Call your Janome dealer and ask to speak with the service person.

SOURCE: The thread is bunching up

Your problem stems from the faulty foot lever mechanism. It's coupled to the upper thread tension control which only provides tension when the presser foot is in the locked down position.

If the presser foot won't positively engage in the down position, there's no upper thread tension and the thread bunches up below. Everything else stems from there. If the needle has hit the bobbin case too frequently then the bobbin timing may also be slightly out and the needle itself will be damaged and won't pass straight through the fabric; it only takes a tiny deflection from the vertical as it passes through the workpiece for the needle to flex and hit the bobbin case.

In summary, your machine is in desperate need of repair and adjustment. You haven't stated which exact Singer model you have, but they are usually easy to get servicing manuals for or you can take the machine to a sewing machine servicing specialist. Most singer spare parts (even for 100 year old machines) are usually still available fairly cheaply from a variety of sources, just enter "Singer spares" into your search engine to find suppliers.

To prevent re-occurrence, clean the mechanism regularly and lubricate it as directed by the manufacturer using sewing machine oil. Depending on how often you use the machine, do (or have done) a more thorough service periodically to check the timing and also that all mechanism screws remain tightly fastened.

I hope that my reply has shed some light on your problem and enables you to decide what to do next. Please take a moment to rate my answer.

SOURCE: I was sewing thick layers and now my needle keeps

By reading your problem I think you have bent the needle shaft as changing the needle does not help. I think you need a repair/service person. Did you use a larger sharp needle when stitching? Top stitch needles are very good, also titanium coated as the tips stay sharper longer.
Sorry I don't have a nicer solution.
Ellie

SOURCE: threads stuck under needle plate

take a tiny screw driver & gently unscrew the screws in the needle plate ,, make sure not to damage the screws & clean out the thread & any lint in that area & also the bobbin area
because if the screws are the slightest bit damaged then when you sew & the fabric moves over them the burrs on the screws will pull the threads in the fabric & can damage the fabric