Do not trifle with the current in the Piscataqua. Make sure your
fuel and filters are good, and you can rely on your
engine. You will NOT be sailing up the river, and if the
engine quits there are no soft spots to hit.
Time your
passage with the flood, from about an hour after low tide until about an hour after high (more or less, depending on where you are.)
The first (Memorial) bridge opens on the hour and half-hour. The next (Sarah Long) bridge is under construction but at this point the
lift span has not been installed, so you get a free pass there.
The set of bridges just before Great Bay Marina (which isn't in Great Bay, btw) are fixed with a clearance of 46' at high tide, with a height of tide normally around 9'. If you can't clear that at most tides, I'd reconsider your choice of
marinas. There's really no place to enjoy even a quick day sail above the bridge, so you'll be passing under it a lot.
I should mention that some folks do manage to push a sailboat against the current, which only hits max (around 6 knots) in a few narrows. The rest of the way the ebb will be 4-5 knots, so if you're up for a very long steam, it can be done. Which of course puts you at the last bridges (the oldest of which is called the General Sullivan Bridge) at low tide, if you need those extra few feet.