Stand at the ledge and press the jump key (default up arrow) while standing still, and you should clear the stalagmites just fine. Continue right.
Three columns of falling rocks block your path. Getting past the rocks at (1) is pretty straightforward. Getting past the other rocks requires you to notice that their timing is slightly off. Wait to advance until rock (3) is falling well after rock (2) falls. You can then run past them successfully.
You'll find a similar solution in this next area. Rocks (1) and (2) should be timed so that the one closest to you drops well before the other. Getting past rock (3) is easy -- just be sure to run as you pass under the tentacles at (4) or else you'll become lunch. Continue right.
It was about this point that I began to remember just how difficult this game was. Argh. OK -- make sure you're running to get past the tentacles at (1) (a jump also works) but stop before you reach the munching bottom-feeder at (2), which you can jump across. Likewise, jump across the bottom-feeders at (3) and (4).
Use your gun to shoot the tentacles, which then pull up enough for you to quickly jump between the bottom-munchers. Be fast, though, as the tentacles return rather quickly.
Jump across the pit then jump over the muncher.
Run through an uneventful frame then blast away this wall. It seems pointless at the moment, but we'll be back.
Now, you want to backtrack left.
Once at this frame, you'll need to continue along the upper ramp. Remember to get past the paired falling rocks by timing your jump so that the closest one is falling well ahead of its partner.
Ditto in this frame as well. Before exiting, shoot at the bat suspended from the ceiling.
Follow the bat left, which is quickly devoured by the tentacles upper-left. You are now free to leap across from stalactite to stalactite. Use the left arrow to reach for each stalactite.
Continue all the way left and drop onto this previously-visited rock.
Walk off the left edge of the rock, then shoot its base. The rock topples over, creating a ramp for you to reach the upper tunnel.
Continue right, leaping over these pits as you go.
Continue right, leaping over this pit. It's a good idea to practice taking these as running leaps, as you'll soon need to jump them at full speed.
Blast the rock supporting the aquifer in this next chamber.
Time to take a running leap. Backtrack left, keeping pace ahead of the rushing torrent.
Continue all the way left, finally being popped up by the cork-like rock you noticed previously.
Continue right. Destroy this wall.
You can continue right or ascend the stairs. My choice is the stairs.
Head left to get your bearings -- you're back at this frame, just on the other side of the bridge.
Continue right, this time jumping over the stairs.
Had you explored this far previously, you would have seen this frame as an impassible waterfall.