The EU made over 65% of laws pertaining to UK, including the immigration laws that forced all EU countries to accept borderless immigration from all EU member countries. Turkey is a member; therefore, huge waves from other parts of the middle east and africa went first to Turkey, then headed for all the best European welfare locations, chiefly Germany, France and mostly London.
The only attempt to 'force' EU countries to accept non-EU citizen immigrants was the Merkel-inspired scheme to disperse Syrian refugees throughout the EU pro rata by population. The UK refused (under the then PM Cameron). Since the UK has never been part of the Schengen passport-free, open-borders zone, which only includes certain EU countries, there was no way it could be enforced without UK consent. (Hence the notorious 'jungle' camps at Calais etc - migrants who had made it across the Schengen area but couldn't pass the UK border.)
Otherwise, immigration to the EU from the third world and all other non-EU countries, whether or not those immigrants arrive via the EU, has been entirely the responsibility of successive UK governments. The UK government, and only the UK government, has controlled UK borders throughout the 37 years of EEC/EC/EU membership.