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The phrase "achieved a bridgehead" is correct and usable in written English.
It is typically used in a military context to describe the establishment of a secure position in enemy territory, but it can also be used metaphorically in business or strategic discussions.
Example: "After months of negotiations, the company finally achieved a bridgehead in the Asian market, allowing them to expand their operations."
Alternatives: "established a foothold" or "gained a position".
Exact(1)
To its north the panzergrenadiers of 3rd SS Panzergrenadier Division Totenkopf achieved a bridgehead across the Psel and tanks had been brought across, but they had yet to take Hill 226.6 and there was a 5 km gap between Totenkopf and Leibstandarte.
Similar(59)
As a result of the withdrawal, Canadian efforts would focus on achieving a bridgehead at San Leonardo.
Achieving a spiritual experience.
Having achieved these objectives, the division was then to create and secure a bridgehead focused around the captured bridges until they linked up with advancing Allied ground forces.
Beginning on 4 December, four infantry divisions one British, one Canadian, one Indian and one New Zealand (which included an armoured brigade)—and two armoured brigades (one British and one Canadian) of the British V and XIII Corps attacked heavily defended German positions along the Moro River, achieving several exploitable bridgeheads by 8 December.
Schools – and particularly female education – are seen as a bridgehead of "corrupting modernity".
It gave Miliband a bridgehead into his wider argument about who will prosper from the recovery.
The osteospermums used Tresco as a bridgehead before launching their invasion of the gardens of Britain.
They have a bridgehead in London and the South East of England.
Soon the Iberian Peninsula, up in arms, became a bridgehead on the Continent for the British.
But he is also now the leader of a party with MEPs in Brussels and a bridgehead in parliament.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com