But the difference is that the parties in the US and the UK aren't very powerful in their own regard. Trump wasn't even a member of the republican party until very recently, and he won the nomination -- and Sanders forced Clinton very far to the left before dropping out.
On the other hand, the leaders of european parties are mostly selected through opaque backroom deals and yet are very powerful (through "party discipline", another throughly undemocratic concept, all but unheard of in UK/US parties). Policy is made by these leaders (or their appointed nominees) meeting each other and negotiating a compromise, and then (except in very rare cases) every MP of the parties in this negotiation defends in the media it as a solid and credible response to the issues faced by the country (despite, often to great hilarity, the compromise being orthogonal to several very firm positions and red lines drawn by that very same politician just days before) and then votes for it.
The mechanism for this power is that few individual MP carry their own seat, they're assigned proportionally, according to the national share of votes won by the party. Party leadership wields significant influence over who runs where (especially who runs in "safe" seats), and thus who gets in. MPs who cause too much trouble will find themselves campaigning in a difficult district, or next to a more popular/well-known candidate.