"I'm looking at my map, and there are two trails that run along the E. Branch Pemi river: The Lincoln Woods Trail to the west, and the Pemi East Side Trail to the east.
According to my map. the Lincoln Woods trail crosses the Franconia Brook, near the junction where it dumps into the East Branch Pemi River. No bridge is marked here.
My map does show the bridges near the Lincoln Woods visitor center and where the Wilderness Trail meets with the East Side trail further up, but I'd rather not go that route."
Not sure what map you are using, but yes there are trails on opposite sides of the East Branch north of the Kanc (Rt. 112). Your "Lincoln Woods Trail" is my "Wilderness Trail" on the west side of the East Branch. There is a substantial bridge on the WT that crosses Franconia Brook where it enters from the north into the East Branch 2.9 miles from the Kanc. The only bridges across the East Branch are next to the Lincoln Woods parking lot on the Kanc and at 5.5 miles upstream, about 0.7 miles above where the Bond Cliff Trail exits north (both are suspension bridges). But, if your intent is to climb Bond Cliff, the only bridges that matter are the first suspension bridge to access the WT (your Lincoln Woods Trail) and the fixed span bridge across Franconia Brook at 2.9 miles; other than those, all streams crossings are smaller without bridges. Good luck; Bond Cliff is one of the finest places to visit in the Whites, IMHO.