Should I be concerned if the IRS Website and TurboTax "can't find my return status"? I filed by mail April 1st...

Your TurboTax account will not reflect anything concerning the mailed tax return.  The software has no way to know when or even if you printed and mailed a tax return.

The IRS is not processing paper tax return at this time.

IRS website - https://www.irs.gov/newsroom/irs-operations-during-covid-19-mission-critical-functions-continue

 The IRS is not currently able to process individual paper tax returns. If you already have filed via paper but it has not yet been processed, do not file a second tax return or write to the IRS to inquire about the status of your return or your economic impact payment. Paper returns will be processed   once processing centers are able to reopen.