I have the opposite experience with cheap, synthetic "moisture wicking" style clothing.
My experience is that it lasts literally forever (assuming I don't tear it) and does not develop an odor problem over time. It actually feels fairly miraculous to me: space-age style comfortable breathable indestructable clothing.
We have active lives here in the NE USA where summers are hot, humid, and tropical. We sweat a looooooot in these clothes: dog walks, tennis, gym wear, yard work, etc. A loooot.
I get the majority of this clothing from Old Navy because they're cheap and sell most items in tall sizes. I've used other cheap brands that are also just as good; I don't think Old Navy is doing anything special in terms of materials but I think they deserve a shout-out for consistency and dedication to extended sizes both large and small.
I stopped buying expensive exercise clothing
many years ago because they only last a season
before they smell too bad
Good news! Absolutely doesn't have to be this way.
Something in your clothes washing routine needs an upgrade.
It may be as simple as cramming less laundry into the washer for each load so that the water:clothing ratio is higher. Another common mistake is using too much laundry detergent. Or maybe the washer is musty.
Assuming it's not either of those things a method that works for me is occasionally doing the "laundry stripping" hot water presoaking method.[1] "Recipes" for this vary and I'm not sure that the commonly recommended triple-ingredient mix is necessary; I suspect a few drops of laundry detergent alone would work just as well. However all of the ingredients are cheap as dirt and this method works so I haven't felt the need to experiment. Note that this isn't specific to synthetic fibers; cotton clothes can have their life multiplied this way as well.
Seems like a lot of work but it really isn't. I have a plastic bin in my basement next to the washer and dryer. I have Load B pre-soaking in the hot water mix while Load A is in the washer. You don't need to do this every time you wash your clothes. Just once every $SOME_OTHER_NUMBER loads.
Vinegar is also really effective in destroying odors. I have saved extremely moldy clothes with vinegar. Vinegar stinks but evaporates fast. If you truly can't stomach the smell (some can't) then tumble dry in the dryer until the vinegar smell is gone.
Also, for goodness' sake, don't toss wet sweaty clothes into the hamper. Let them air dry somehow first.
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[1] https://www.google.com/search?q=laundry+stripping