You can either install a wall hung cabinet with space below for baskets or a floor supported cabinet for storage depending on the style of the items in your bathroom.

Over the toilet and on the wall opposite the toilet. This is why double sinks are a bad.

I do not like dirty laundry in my bathroom and a nice hotel type wall mounted rack can let you add some towels easily I need an actual picture oof the whole space to really help. I have found most people have way too much stuff in bathrooms that could easily be stored someehwere else Post the pics here in comments DO NOT start another post.

I'm starting to see why double sinks aren't that great but at the same time, it is a bit of luxury and kind of worked with the space. I think it would have looked odd with a single sink.

Above the vanity basins is just Ikea mirrors. No space for storage as there's a window extremely close to the left mirror.

Leaning towards the baskets under the vanity though I assume I'll have to move them often when I clean.

If it were me, I would give up the towel bar for the cabinet ideas given above -

Would agree, then no space for towels...

What is adjacent to the bathroom walls can you break into that space and install a cabinet? I would see if a 12" deep cabinet could fit into wall across from the toilet. Worst case build some niche into that wall and do cute baskets

Double sinks are a luxury when 2 people use the sinks at the same time several times a week. In your case, you wanted something that should look great but it’s not functional, with stuff stored on the countertop and beneath the vanity, so it’s not even a luxury. Some hotel bathrooms are like that. I guess they are easy to clean once the guest moves out.

I hung a kitchen-type cabinet over my toilet. Only thing if you do that make sure you have room to scrub out the top toilet tank, so plenty of clearance. Then, before you rummage in the cabinet, shut the toilet lid just in case something decides to jump into the toilet. You can also buy little plastic baskets at dollar tree (take a tape measure in your purse). When you need something, bring the basket down, get what you need, and put the basket back. With kids, that's definitely a good plan.

Can you put the hamper somewhere in your bedroom? Then you could use that wall for a cabinet above the towel rods.

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Alternatively, you could switch from a bar to hooks, allowing for a tall cabinet, or shift the towels over and add a corner organizer for more storage.

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Also also, you could put a slimmer cabinet over the toilet...

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You could also replace the Ikea mirrors with Ikea medicine cabinets, which hold a lot, freeing up the drawers for larger things... As for under the vanity, if you have to put stuff there, get some white plastic bins with lids -- Ikea has them in lots of sizes -- which will keep the water and dust out.

If you use baskets under the vanity, use large ones so there are only 2 things to move when cleaning. Look for baskets on casters so they are easy to move. And that will keep them off the floor and out of any spilled water.

If you use a cabinet across from the toilet, add hooks to the side or even front to hang towels.

I am not a fan of laundry baskets in bathrooms because of the humidity. Perpetually damp clothes in a pile. Ick.

If you use baskets under the sink, they should be lined and lidded. Everything that you sweep up from the floor can also get into the baskets. And put glides or something under them, so they are easy to slide out. What would you keep in them?