It takes 5 minutes for the water to get hot in our bathroom shower and sink which are only 30-feet (3/4 inch copper piping to bath) from our 50-gallon gas-fired hot water heater. We waste so much water!

Looked into recirculation pump with dedicated return line. Plumber said he couldn’t understand why I’m having this problem when bathroom is so close. Is there a simple solution to my problem?

I'd start with the recirculation pump. Is it running? Is the check valve clogged? What's looped into the bathroom branch? You're pulling that coldwater out of someplace. It can't be the shower mixing valve if the lavatory's affected also. Sounds like a cross connection. Any work done lately? Tom

Don't have a recirc pump. I was considering putting one in to solve my problem. But plumber said he didn't understand why I was having the problem in the first place.

Then have your plumber,( or you could do this.) start to shut off or disable the mixing valves(the one handled ones) one at a time and run hot water to the bathroom after each one to see if there's a cross over in one of them. If you haven't a recirculating system then that's where I'd start. Eliminating them one at a time. Ya got to start someplace. Looking at it and scratching your head ain't going to get it. Good luck and let me know what you find. Tom