How can I secure expandable shower rods to the bathroom wall?

What about double stick tape? Or the sticky pad that is used for command hooks.

Ceiling mounted curtain rods and hooks are another option.

https://www.homedit.com/install-ceiling-mounted-shower-curtain/

Closet rod/pole hanging sockets come in stainless steel, plastic, painted steel -

Closet rod pole sockets. Get the cheap plastic ones and purchase two sets. Use the open socket from each set. You might have to cut the sides down or if your rod is thick or you might have to remove the sides completely. All you really need is about an inch of plastic left at the bottom. These at Home Depot are $1.47.

https://www.homedepot.com/p/Everbilt-1-3-8-in-White-Plastic-Closet-Pole-Sockets-2-Pack-HD-0037-25/205561142

Hi Brenda, here are some great ideas to keep that shower rod in place

https://www.rotatorrod.com/blogs/bathroom-bliss/tension-shower-rod-fall-down#:~:text=You%20can%20fix%20that%20by,of%20the%20caps%20with%20scissors.

Mine has rubber ends, and directions on how to expand it. Just twist to make it tight and you will be good.