The phrase "a bridgehead across" is correct and usable in written English.
It can be used in contexts related to military strategy, metaphorical discussions about establishing a foothold in a new area, or in business when discussing market entry.
Example: "The company established a bridgehead across the European market, allowing it to expand its operations significantly."
Alternatives: "a foothold in" or "an entry point into".
Their first move was to be securing a bridgehead across the Yser (Ijzer) canal.
He retreated in good order, however, with only limited losses, and a bridgehead across the Hellespont was established.
The more lightly armed marines have also been slowed by fighting in the city of Nasiriya, near a bridgehead across the Euphrates that is critical for keeping the supply lines open.
Shea's infantry division was to advance to establish a bridgehead across the Nahr el Faliq and create a gap in the front line for the cavalry.
The River Touques has two branches running through the town, and the 13th Parachute Battalion were given the objective of establishing a bridgehead across the river.
On 14 April, the 9th Panzer Division established a bridgehead across the Haliacmon river, but an attempt to advance beyond this point was stopped by intense Allied fire.
It was also about using the games as a bridgehead into initiating creativity across all our communities.
A pontoon bridge was floated across the Santi Petri creek and a battalion sent across to establish a bridgehead prior to the arrival of the main force.
Alexander threw his cavalry across the river to gain a bridgehead, and the Persians naturally countered with their cavalry, so that a piecemeal engagement followed.
At 19 00 that evening, following the clearance of nearby Pakchon by D Company 3 RAR, the Australians sent two companies across the river to establish a bridgehead and prevent the North Koreans from consolidating their position on the western bank.
On 31 March 1945, the 100th Infantry Division crossed the Rhine and moved south in the wake of the 10th Armored Division and then east across the Neckar River, establishing and enlarging a bridgehead from 4 to 11 April.
