Using a herbicide on the bindweed is rarely that effective as the root system is far too extensive. I know you don't want to put off laying down turf so I can make a suggestion for control, NOT eradication. I don't know how large you want your lawn so the amount of work may not appeal to you.
I wasn't making a lawn but a large flower bed. I cleared weeds and stoloniferous grass along with as much bindweed and its roots as I could. I then used a garden fork to repeatedly dig down and loosen the soil from more roots that I found. I went down around 3 feet and took out every piece I could find. I really went through all the soil! Because I hadn't tilled or chopped the soil up, most of the roots were intact. Their length and branching was unreal. If you've tilled your soil, your task is more difficult as a new plant can grow from each piece of cut root, even if it's an inch long. While bindweed roots can go very deep, most of them are found in the top 2 feet if soil (which is why I went down 3 feet to play it safe).
Once you've cleared the soil of most (you'll likely think ALL) roots, you can lay your turf. But you'll have to be vigilant to look for any new bindweed attempting to grow from missed roots. If you cut all new leaves formed as soon as you see them, you'll eventually starve out the food stores from any remaining roots. But you can't allow any to grow or food stores will be built up in the roots again and your task will ake much longer. I haven't needed to do this but I've read it can take 2-5 years before no more plants will emerge - unless there are already bindweed seeds in the soil. It's a prolific seeder as most weeds are.
I'm sorry to say but no one method is foolproof. Some feel a herbicide (which I won't use) along with persistent removal of top growth works best. It's still going to take a year or two even using both before your lawn is bindweed-free.