Try running water down through your downspout.
If a simple hose spray doesn’t work, create a seal using a rag or hand towel at the top of the gutter, then spray your garden hose into the top of the downspout.   Break out the power tools. A leaf blower, shop vac, or even power washer can provide enough force to unclog your downspout.   Hire a professional. If all else fails, call in the professionals who have the specialized tools and expertise to clean and fortify your gutter and downspouts.