Its snapshot survey of 900 heads found 90% were struggling to recruit.
The government says it is investing Â£1.3bn up to 2020 to attract new teachers where they are most needed.
But three-quarters of respondents to the survey said it was now tougher to recruit than 12 months ago.
And nearly nine out of 10 said the situation was creating extra work and stress for staff.
While some 70% were using more support staff, about a fifth had worked with other schools to share or recruit teachers.
Some 41% of those surveyed had arranged for more lessons to be taught by the head or deputy head.
The biggest areas of subject shortage were in the key subjects of maths, science and English.
Some schools faced difficulties in attracting languages, geography and history teachers.
ASCL general secretary Malcolm Trobe said: "Teacher shortages leave schools with no option other than to use stop-gap solutions.
"Schools have to put teachers in front of classes.
"If they cannot recruit the staff they need, this means using supply staff and non-specialists to cover the gaps.
"While these staff often do a very good job in difficult circumstances, it is no substitute for having permanent teachers who are experts in their subjects.
"Without this supply of teachers, there is a danger that some of the progress which has been made will be lost.
"It will certainly be extremely difficult if not impossible to raise standards further."
He said schools that struggled to find enough maths or science teachers, for example, would cut down on the number of ability groups they offered.
"This is being done because schools cannot find enough teachers not because it is desirable," said Mr Trobe.
He urged the government to do more to make teaching a more attractive profession.
A Department for Education spokesman said: "The government is investing hundreds of millions in teacher recruitment, offering generous bursaries and scholarships in Stem [science, technology, engineering and mathematics] subjects, plus backing schemes like Teach First and the National Teaching Service to get great teachers where they're most needed - and... we've given schools unprecedented freedom over staff pay, allowing them to attract the brightest and the best.
"We are committed to raising the status of teaching and want to work with ASCL and the profession to make it an attractive career choice."
The survey comes after the government missed its teacher recruitment targets in England for the fourth year in a row, leading to shortfalls in many subjects.
This followed a shake-up of teacher training in which the numbers to be recruited on traditional post-graduate routes were initially reduced as a new school-based training scheme was introduced.

Daniel Jones, 59, from Enfield, is currently serving a prison sentence for his role in the infamous £25m jewellery heist of 2015.
He has now admitted being involved in an attempt to break into a safe in Chatila jewellers in Old Bond Street, Mayfair, in August 2010.
Two other men have pleaded not guilty.
Terry Perkins, 68, of Enfield, has denied making off with goods worth more than a million US dollars from a jewellers between 27 and 31 August 2010.
Charles Matthews, 54, from Virginia Water, Surrey, denies one count of handling stolen goods.
They are both due to stand trial on 27 February at Southwark Crown Court.
Head teachers in England are having to share staff with other schools as they try to find "stop-gap solutions" to fill vacancies, the Association of School and College Leaders says.
[NEXT_CONCEPT]
One of the ringleaders behind the Hatton Garden raid has admitted trying to steal £1m of gems from a London jewellers five years earlier.