Just letting the dog out is NOT enough exercise for your dog; no matter what size your yard is. The same goes for small dogs.
Crating is very helpful BUT only when used in conjunction with a very consistent schedule of feeding and LONG, BRISK walks.
I've taken in many rescue dogs in bad shape and the crate is an interim tool only. After a few weeks, you should be able to leave the door open and let the dog use the crate as a den if it is large enough.
If it takes longer; you need to adjust what you're doing with the walks/feeding schedules until it does work.
How do you know if the walk is long enough? Your heart rate should go up a bit AND the dog will probably have to poop at least once along the way. If he doesn't; keep on walking until he has to go. You'll learn what you need to do quickly if you pay attention.
It will become like clockwork. You'll be able to bet on how far you have to walk before he goes without fail.
Ideally do a long walk (at least 25-45 mins) every morning at the same time. If you need to take your coffee with you; that's what you need to do. The dog will probably go within 5-10 mins but they (and you) need the exercise.
You can NOT train a dog to "be good" with less exercise. They need it and the more they get, the better their behavior will be.
Sorry if you like to sleep in on certain days. You need to be consistent. Do the same in the evening after the dog eats.
Your dog needs to have some snacks during the day in addition to the main evening meal. Plan them accordingly with your walks.
At night, cut the water a little bit when it gets late. Always keep ample water out during the day and give fresh water frequently.
If you drank a Super Big Gulp before bed; you'd have to pee at least once during the night too. :)
I've found that most dogs respond very, very quickly to a consistent schedule and the long, brisk walks; no matter what the age or background.
Good luck and remember...if it doesn;t work; YOU need to change what you're doing until it does work. With prolonged training problems; it's usually that the dog has trained you instead of you training the dog.