The state I live in, South Australia, banned lightweight plastic shopping bags some time ago. The absence of these bags littering roadsides and parks has been wonderful. 
    
I feared we would start seeing the cheap, heavier reusable bags turning up in their place, but that hasn't happened yet thankfully.. and hopefully never will.
    
Unfortunately, there has been an increase in sales of kitchen bin liners - which is what many people used to use the free lightweight bags for - so there needs to be an awareness campaign about that issue and pressure brought upon manufacturers of bin liners to green up their act a little.
   
One of the other challenges for many people (myself included initially) is forgetting to take reusable bags when shopping.

While some supermarkets provide free boxes, they aren't always available and you then need to buy more bags (I've been guilty of this too). Before long, you can have dozens of reusable bags floating around your house.

Depending on the type of reusable bags purchased, these could ultimately wind up in landfill; which somewhat defeats the purpose of the ban of the lighter bags.
   
Initiatives like the Bagshare Project in the USA help address this problem.
 
The project, started by Leni Fried, seeks to help folks who forget to take along their reusable bags when shopping by providing free reusable bags on a loan basis.

Volunteers sew the cloth bags and an ID tag is displayed on each one stating it is a 'Use and Return' bag. When someone forgets their own bags, they can use these and then return them to any BagShare location.
 
The project isn't just for places where lightweight bag bans are in place. The primary aim of BagShare is to transition stores to going disposable bag free.
 
The concept does require a honesty and consideration on the part of the people borrowing the bags of course - to return them in a timely manner ( and in good condition); but I think it's a really nice idea.

Aside from the "green" aspect, it also has a community-building angle and gets people involved in volunteerism - and what wonderful folks these volunteers are to donate their time in such a way!
  
Over 10,000 of these bags have been sewn by volunteers over the last 4 years.
  
Leni invites people to start their own Bagshare project and will provide advice in getting set up. If you would like to support the initiative in other ways, she welcomes donations of sewing machines, fabric, thread and other related materials; as well as pre-made bags that can be used in their current project locations.