How I can prevent Safari popups to increase database size on iPad?
There is no easy way to circumvent this behavior in Mobile Safari.

I know of no alternative browser that accomplishes what you ask, but perhaps if someone else does they will answer or edit my answer.

You can avoid those sites, or give more space when asked. It's one of those "we can't adjust the wind, but we can adjust the sails or steer the boat elsewhere" kind of things at present.