(CNN) About half of Democrats and Democratic-leaning independents want the government to provide a national health insurance program, but don't want it to completely replace private insurance, according to a new CNN poll conducted by SSRS out Monday.

Yet, the candidate who is seen as the strongest on health care is Vermont Sen. Bernie Sanders, the candidate who unapologetically backs "Medicare for All" and wants to see the end of the private insurance industry. When asked which Democratic candidate for president can best handle health care, Democrats and Democratic-leaning independents supported Sanders above the rest (26%).

Following Sanders on health care is former Vice President Joe Biden (18%), Massachusetts Sen. Elizabeth Warren (16%) and California Sen. Kamala Harris (10%). This poll was conducted after the debates last week , in which candidates were asked if they would eliminate private insurance for a national health plan. Most candidates did not raise their hands, though some -- like Warren and Sanders -- did.

Potential Democratic voters' trust in Sanders on health care seems to hold regardless of whether they want to eliminate private insurance. Among those who favor a national plan and want to completely repeal private insurance, 36% said Sanders would do the best job of handling health care.

Those who favor a national plan but still want some private insurance think Sanders would do the best job in slightly lower numbers, but he still tops the pack (27%). Harris gets slightly better ratings on handling health care in this group -- 13% versus only 3% among those who want to completely replace private insurance.

Three in 10 Democrats and Democratic-leaning voters said they favor a national plan and want it to completely replace private insurance. Only 13% of Democrats and 40% overall opposed a national government-run plan.

The total population also slightly favored not replacing private plans (32%), over those who favored a government plan and wanted a total overhaul (21%).

More than eight in 10 potential Democratic voters said they favor a national health insurance plan provided by the government, but just three in 10 favor one which completely replaces private insurance. Overall, 56% of Americans favor a government-provided national health insurance plan, 40% oppose it.

And as among Democrats, more Americans generally say they favor a plan which would not completely replace private insurance (32%), over one which would eliminate them (21%).

Three-quarters of Americans said they feel they understand what candidates mean when they use the phrase Medicare for All, while a quarter said they don't totally get it. Democrats are more likely than Republicans and independents to believe they understand Medicare for All (80% versus 71% for independents and 73% among Republicans).

Most Americans also don't want government provided health insurance to be available to undocumented immigrants living in the US, while only 38% do. Most Democrats want undocumented immigrants to have access to public insurance (66%), while independents and Republicans are mostly against it (34% and 10%, respectively, said it should not be provided). All 10 Democrats on the second night of debates raised their hands when asked whether their health plans would provide coverage for undocumented immigrants.

The CNN Poll was conducted by SSRS June 28 through 30 among a random national sample of 1,613 adults reached on landlines or cellphones by a live interviewer. Results for the full sample have a margin of sampling error of plus or minus 3.0 percentage points. It is larger for subgroups.