The current rate of suicide is "unacceptable", leading MPs have said, as they called for GPs to receive better training in assessing a patient's risk.
The Health Select Committee said that it had heard evidence that clinicians sometimes fail to recognise patients who are suicidal.
It called on the General Medical Council, Royal College of GPs and Health Education England to improve training for students and clinicians in the assessment of suicide risk.
Around a third of people who end their lives by suicide are in contact with their GP before their death, but are not receiving specialist mental health services, MPs said.
Meanwhile, the committee said it was "unacceptable" that only 60% of people who present at emergency departments for self-harm receive a psycho-social assessment when self-harm is the single biggest indicator of suicide risk.
And MPs expressed disappointment that the Government had not acted on a previous recommendation calling for all patients who are discharged from inpatient care to receive follow-up checks within three days.
"We remain concerned that the ongoing workforce shortfall is the key barrier to this goal," they wrote.
The MPs said that the Government's suicide prevention strategy needs greater focus on implementation.
Chairwoman of the Health Committee, Dr Sarah Wollaston MP, said: "The clear message we have heard throughout our inquiry is that suicide is preventable.
"The current rate of suicide is unacceptable and is likely to under-represent the true scale of the loss of life.
"If the Government wishes to be truly ambitious in reducing the toll of suicide, there are many further steps which it could take, which we have set out in this report. The Government must prioritise effective implementation of its strategy because without it, any strategy is of very limited value."
In 2015, there were 4,820 people are recorded as having died by suicide in England, but MPs said the figure is likely to be "much higher".
Commenting on the report, Izzi Seccombe, chairwoman of the Local Government Association's Community and Wellbeing Board, said that suicide was a "major national public health concern".
She added: "While the overwhelming majority of councils now have a suicide prevention plan in place, even though there is no requirement for them to do so, we can only really tackle the issue alongside other public and private organisations, such as schools, railway operators, supermarkets, hospitals and police stations.
"Suicide is preventable, but it needs to be everybody's business to work together to tackle this tragic loss of life."
Ian Hulatt, professional lead for mental health at the Royal College of Nursing, said: "The report exposes the gap between Government rhetoric and action.
"The funding is not getting through and little is known about the quality of local plans or their ability to reach the people who need support."
Dr Peter Aitken, spokesman for the Royal College of Psychiatrists, added: "A shortage of liaison psychiatrists, who work in A&E at the point of mental health crises, is a huge barrier to rolling out liaison services in every acute hospital in England.
"Liaison psychiatry is effective and mental health staff save lives. We echo calls that the Government must ensure there are enough well-trained staff to establish and sustain liaison psychiatry services."
Health Secretary Jeremy Hunt said: "We agree with the committee that every death by suicide is a tragedy which has a devastating effect on families and communities.
"We are taking action to ensure we quality-assure every local plan so they reach the most vulnerable people - and local authorities now are supported by record mental health funding, with big developments in the availability and quality of crisis care services as a result."
Javed Khan, chief executive of the children's charity Barnardo's, said: "Tackling this issue is everyone's responsibility and we agree much more needs to be done to reduce the suicide rate.
"The care leavers we work with tell us mental health is a huge issue for them and they often feel abandoned, without any proper mental health support, and they often have no-one to turn to.
"Many say they are unable to access clinical services so we want the Government to commit to spending some of the £1.4 billion promised for children's mental health to help this vulnerable group of young people."