In my experience, a four-fold decrease is not typical. But there are many variables in play, some you have not addressed. Is your positioning the same? Ad position has an effect on conversion rates.
There's also impression share. I would want 100% or close to it. By halving your budget, chances are your IS is half or even less. This is a very important metric that most don't know about or ignore. You see, if you give competitors a bigger share of the ad inventory, they are the ones more likely to get the conversion. If you show your ads half as much, you are giving them those conversions and that seems to be what is happening here. So increase your budget. You obviously are doing well so I don't know why you decreased it in the first place.
You should also look into the auction insights report and see who your real competitors are. I'd be willing to bet you are losing your conversions to those with high impression shares. They are showing more often so the times you didn't, your clients simply went there because they didn't see your alternative. It sounds too that people may be searching multiple times and clicking many ads when they do. By seeing your competitors' more than once, they are in effect being sold twice by them whereas you are doing it just once and that is affecting your conversion rate (you are not top of mind since they land on your page just once), another reason to increase your budget. In fact if your IS was not in the 90% range, I'd increase the budget higher than you had it before. Your conversion rate may be higher still. At the very least, even if it's 10%, you'd have more leads each day because you'd get more clicks and taking some away from your competition.
